5993:
7497:
5517:
7121:
38:
8774:
5988:{\displaystyle x={\frac {\,{\begin{vmatrix}5&3&-2\\7&5&6\\8&4&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;y={\frac {\,{\begin{vmatrix}1&5&-2\\3&7&6\\2&8&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;z={\frac {\,{\begin{vmatrix}1&3&5\\3&5&7\\2&4&8\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}}.}
2510:, and the solution set is the intersection of these planes. Thus the solution set may be a plane, a line, a single point, or the empty set. For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is infinite and consists in all the line passing through these points.
2315:
7492:{\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;=\;&&&0\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;=\;&&&0\\&&&&&&&&&&\vdots \;\ &&&\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;=\;&&&0.\\\end{alignedat}}}
2691:
2545:
95:
9038:
1970:
2332:
2983:
1837:
2624:
8600:
2617:
2610:
2962:
3265:
2310:{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}.}
3631:
5506:
1521:
6723:), computers are often used for larger systems. The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as
1291:
670:
2820:
3132:
3334:
if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. It follows that two
6005:
Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.) Further, Cramer's rule has very poor numerical properties,
2649:
It must be kept in mind that the pictures above show only the most common case (the general case). It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines
3317:
is the difference between the number of variables and the rank; hence in such a case there is an infinitude of solutions. The rank of a system of equations (that is, the rank of the augmented matrix) can never be higher than + 1, which means that a system with any number of equations can always be
2967:
are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single
2682:
if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as
3500:
6778:
A completely different approach is often taken for very large systems, which would otherwise take too much time or memory. The idea is to start with an initial approximation to the solution (which does not have to be accurate at all), and to change this approximation in several steps to bring it
6634:
5319:
1832:{\displaystyle x_{1}{\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix}}+x_{2}{\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix}}+\dots +x_{n}{\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}
5266:
976:
7834:
3868:
3289:
In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent.
1860:, and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. In any event, the span has a
2645:
The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point.
3359:
When the solution set is finite, it is reduced to a single element. In this case, the unique solution is described by a sequence of equations whose left-hand sides are the names of the unknowns and right-hand sides are the corresponding values, for example
2552:
In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns. Here, "in general" means that a different behavior may occur for specific values of the coefficients of the equations.
564:
7739:
2957:{\displaystyle {\begin{alignedat}{5}x&&\;-\;&&2y&&\;=\;&&-1&\\3x&&\;+\;&&5y&&\;=\;&&8&\\4x&&\;+\;&&3y&&\;=\;&&7&\end{alignedat}}}
3260:{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&y&&\;=\;&&1&\\2x&&\;+\;&&y&&\;=\;&&1&\\3x&&\;+\;&&2y&&\;=\;&&3&\end{alignedat}}}
261:
4260:
3626:{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\end{alignedat}}}
7001:
3309:. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has
6419:
If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which
6279:
6487:
4038:
7126:
5324:
3505:
3137:
4173:
5501:{\displaystyle {\begin{alignedat}{7}x&\;+&\;3y&\;-&\;2z&\;=&\;5\\3x&\;+&\;5y&\;+&\;6z&\;=&\;7\\2x&\;+&\;4y&\;+&\;3z&\;=&\;8\end{alignedat}}}
5292:. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the difference lies in how the computations are written down.
3711:
4536:
3106:
2806:
7603:) then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties:
1286:{\displaystyle {\begin{cases}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\dots +a_{mn}x_{n}=b_{m},\end{cases}}}
839:
7767:
4541:
4193:
3762:
Different choices for the free variables may lead to different descriptions of the same solution set. For example, the solution to the above equations can alternatively be described as follows:
6417:
3768:
6111:
2811:
are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations.
1415:
7536:
6371:
6045:
1931:
6338:
2425:
1474:
1353:
7084:
2825:
761:
569:
7041:
6888:
3419:
665:{\displaystyle {\begin{alignedat}{5}2x&&\;+\;&&3y&&\;=\;&&6&\\4x&&\;+\;&&9y&&\;=\;&&15&.\end{alignedat}}}
358:
4348:
941:
4479:
3755:
of the solution set. For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter
8273:
6703:
6681:
6659:
6304:
3470:
286:
to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the
4089:
7685:
6479:
6144:
6449:
6199:
545:
513:
6851:
888:
6825:
6805:
908:
862:
713:
693:
17:
132:
4188:
6896:
6779:
closer to the true solution. Once the approximation is sufficiently accurate, this is taken to be the solution to the system. This leads to the class of
6207:
7094:
6629:{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} +\left(I-A^{-1}A\right)\mathbf {w} =A^{-1}\mathbf {b} +\left(I-I\right)\mathbf {w} =A^{-1}\mathbf {b} }
6783:. For some sparse matrices, the introduction of randomness improves the speed of the iterative methods. One example of an iterative method is the
6006:
making it unsuitable for solving even small systems reliably, unless the operations are performed in rational arithmetic with unbounded precision.
3916:
2557:
In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an
3126:
It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations
1854:) to be brought to bear. For example, the collection of all possible linear combinations of the vectors on the left-hand side is called their
4519:
Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.
4097:
3893:
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows:
8604:
8160:
3642:
3111:
are inconsistent. In fact, by subtracting the first equation from the second one and multiplying both sides of the result by 1/6, we get
5261:{\displaystyle {\begin{aligned}\left&\sim \left\sim \left\sim \left\\&\sim \left\sim \left\sim \left\sim \left.\end{aligned}}}
8632:
7679:
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
3039:
2739:
8965:
1903:
9023:
3900:
Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.
8498:
8488:
8477:
8415:
8299:
7829:{\displaystyle \left\{\mathbf {p} +\mathbf {v} :\mathbf {v} {\text{ is any solution to }}A\mathbf {x} =\mathbf {0} \right\}.}
6752:
has some special structure, this can be exploited to obtain faster or more accurate algorithms. For instance, systems with a
2977:
6310:×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using
6002:, while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms.
780:
6714:
5175:
5094:
5010:
4923:
4823:
4727:
4637:
4549:
4390:
6737:. This is mostly an organizational tool, but it is much quicker if one has to solve several systems with the same matrix
3863:{\displaystyle y=-{\frac {3}{7}}x+{\frac {11}{7}}\;\;\;\;{\text{and}}\;\;\;\;z=-{\frac {1}{7}}x-{\frac {1}{7}}{\text{.}}}
943:. This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on
7960:
5309:
is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two
6376:
9072:
8521:
8458:
8433:
8343:
8321:
8281:
8255:
8229:
7106:
6073:
81:
59:
52:
9013:
8975:
8911:
1358:
7508:
6343:
6017:
3286:. Any two of these equations have a common solution. The same phenomenon can occur for any number of equations.
8313:
6313:
8561:
Peng, Richard; Vempala, Santosh S. (2024). "Solving Sparse Linear
Systems Faster than Matrix Multiplication".
8352:
Harrow, Aram W.; Hassidim, Avinatan; Lloyd, Seth (2009), "Quantum
Algorithm for Linear Systems of Equations",
8136:
2371:
1888:
independent vectors a solution is guaranteed regardless of the right-hand side, and otherwise not guaranteed.
1420:
1299:
8105:
3294:
8753:
8625:
7591:) solution, which is obtained by assigning the value of zero to each of the variables. If the system has a
7046:
4527:
3475:
To describe a set with an infinite number of solutions, typically some of the variables are designated as
2567:
In general, a system with more equations than unknowns has no solution. Such a system is also known as an
721:
8858:
8708:
7955:
7009:
6856:
2488:
455:
3363:
292:
8657:
7922:
6890:
is used at the start of the algorithm. Each subsequent guess is computed using the iterative equation:
4288:
1872:
463:
913:
9003:
8652:
6167:
4485:
4381:
2492:
6661:
has completely dropped out of the solution, leaving only a single solution. In other cases, though,
4106:
3925:
985:
141:
8995:
8878:
8247:
7999:
7945:
6756:
4522:
There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are
4489:
2491:. Because a solution to a linear system must satisfy all of the equations, the solution set is the
372:
46:
6686:
6664:
6642:
6287:
3432:
9067:
9041:
8970:
8748:
8618:
7981:
2564:
In general, a system with the same number of equations and unknowns has a single unique solution.
2507:
8805:
8738:
8728:
7986:
7853:
7734:{\displaystyle A\mathbf {x} =\mathbf {b} \qquad {\text{and}}\qquad A\mathbf {x} =\mathbf {0} .}
6760:
5999:
4050:
2558:
2364:
1870:
vectors that do guarantee exactly one expression; and the number of vectors in that basis (its
1862:
963:
767:
282:
63:
8146:
8820:
8815:
8810:
8743:
8688:
8115:
7975:
7906:
3298:
2568:
2319:
1856:
6454:
6119:
8830:
8795:
8782:
8673:
8371:
8221:
7965:
7927:
7641:
7616:
6427:
6177:
4523:
4506:
4359:
3319:
2653:
A system of linear equations behave differently from the general case if the equations are
1897:
1867:
1850:
524:
489:
8:
9008:
8888:
8863:
8713:
8242:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
8030:
7902:
7592:
6830:
3120:
2684:
2655:
2522:
2503:
1496:
867:
439:
435:
412:
99:
8375:
6729:. Secondly, the algorithm does not exactly do Gaussian elimination, but it computes the
4530:. The following computation shows Gauss–Jordan elimination applied to the matrix above:
3759:. An infinite solution of higher order may describe a plane, or higher-dimensional set.
558:
The simplest kind of nontrivial linear system involves two equations and two variables:
466:. For an example of a more exotic structure to which linear algebra can be applied, see
256:{\displaystyle {\begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}}
9062:
8718:
8570:
8395:
8361:
8240:
7874:
6810:
6790:
6764:
3422:
3306:
2480:
1512:
944:
893:
847:
698:
678:
447:
408:
8504:
8079:
4255:{\displaystyle {\begin{aligned}3z+2={\tfrac {7}{2}}z+1\\\Rightarrow z=2\end{aligned}}}
3487:), meaning that they are allowed to take any value, while the remaining variables are
675:
One method for solving such a system is as follows. First, solve the top equation for
8916:
8873:
8800:
8693:
8517:
8494:
8473:
8454:
8429:
8411:
8387:
8339:
8317:
8295:
8277:
8266:
8251:
8225:
3748:
2428:
467:
6996:{\displaystyle {\mathbf {x}}^{(k+1)}=D^{-1}({\mathbf {b}}-(L+U){\mathbf {x}}^{(k)})}
8921:
8825:
8678:
8580:
8399:
8383:
8379:
7970:
6780:
6753:
6730:
5301:
4373:
3348:
3302:
396:
388:
8187:
459:
8980:
8773:
8733:
8723:
8165:
7870:
7664:
7656:
6768:
6683:
remains and hence an infinitude of potential values of the free parameter vector
3906:
Solve this equation, and then back-substitute until the entire solution is found.
2529:
1492:
443:
123:
119:
6274:{\displaystyle \mathbf {x} =A^{+}\mathbf {b} +\left(I-A^{+}A\right)\mathbf {w} }
2600:
The following pictures illustrate this trichotomy in the case of two variables:
8985:
8906:
8641:
8549:
7940:(the free standard package to solve linear equations numerically; available in
6147:
3297:, any system of equations (overdetermined or otherwise) is inconsistent if the
1484:
431:
416:
364:
6047:, the entire solution set can also be expressed in matrix form. If the matrix
3732:. Any point in the solution set can be obtained by first choosing a value for
3335:
linear systems are equivalent if and only if they have the same solution set.
9056:
9018:
8941:
8901:
8868:
8848:
8335:
7663:. In particular, the solution set to a homogeneous system is the same as the
6784:
6772:
6771:. Special methods exist also for matrices with many zero elements (so-called
6724:
3903:
Repeat steps 1 and 2 until the system is reduced to a single linear equation.
3897:
In the first equation, solve for one of the variables in terms of the others.
3636:
The solution set to this system can be described by the following equations:
3023:
2548:
The solution set for two equations in three variables is, in general, a line.
1950:
1508:
404:
400:
6719:
While systems of three or four equations can be readily solved by hand (see
8951:
8840:
8790:
8683:
8391:
7932:
4033:{\displaystyle {\begin{cases}x+3y-2z=5\\3x+5y+6z=7\\2x+4y+3z=8\end{cases}}}
3426:
2690:
2433:
1844:
442:, other theories have been developed. For coefficients and solutions in an
434:, but the theory and algorithms apply to coefficients and solutions in any
5271:
The last matrix is in reduced row echelon form, and represents the system
8931:
8896:
8853:
8698:
7568:
5310:
1480:
967:
427:
423:
376:
107:
2544:
94:
8960:
8703:
7655:
These are exactly the properties required for the solution set to be a
2518:
6067:
rows are independent), then the system has a unique solution given by
4168:{\displaystyle {\begin{cases}y=3z+2\\y={\tfrac {7}{2}}z+1\end{cases}}}
2528:. The solution set is the intersection of these hyperplanes, and is a
2317:
The number of vectors in a basis for the span is now expressed as the
550:
However, most interesting linear systems have at least two equations.
8758:
6720:
3752:
3344:
2576:
2496:
2331:
392:
384:
368:
8584:
7619:
representing solutions to a homogeneous system, then the vector sum
7502:
A homogeneous system is equivalent to a matrix equation of the form
8926:
8575:
3706:{\displaystyle x=-7z-1\;\;\;\;{\text{and}}\;\;\;\;y=3z+2{\text{.}}}
2495:
of these lines, and is hence either a line, a single point, or the
8366:
8126:
8124:
8060:
3026:
from the equations, that may always be rewritten as the statement
1507:
One extremely helpful view is that each unknown is a weight for a
8610:
7941:
7897:
has at least one solution. This occurs if and only if the vector
7636:
is a vector representing a solution to a homogeneous system, and
7583:
Every homogeneous system has at least one solution, known as the
2982:
1488:
451:
380:
3421:. When an order on the unknowns has been fixed, for example the
3270:
are inconsistent. Adding the first two equations together gives
2368:
of a linear system is an assignment of values to the variables
8936:
8599:
8194:
8161:"New Algorithm Breaks Speed Limit for Solving Linear Equations"
8121:
8048:
7937:
4091:, and plugging this into the second and third equation yields
3022:. When the system is inconsistent, it is possible to derive a
2623:
2440:
A linear system may behave in any one of three possible ways:
1884:, but it can be smaller. This is important because if we have
1495:
are also seen, as are polynomials and elements of an abstract
844:
This results in a single equation involving only the variable
98:
A linear system in three variables determines a collection of
7949:
6306:
is a vector of free parameters that ranges over all possible
5998:
For each variable, the denominator is the determinant of the
287:
2616:
4161:
4026:
3282:, which can be subtracted from the third equation to yield
3101:{\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;3x+2y=12}
2801:{\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;6x+4y=12}
2609:
1279:
462:. For finding the "best" integer solutions among many, see
458:. For coefficients and solutions that are polynomials, see
367:, a subject used in most modern mathematics. Computational
249:
8493:. Society for Industrial and Applied Mathematics (SIAM).
4285:= −15. Therefore, the solution set is the ordered triple
27:
Several equations of degree 1 to be solved simultaneously
7873:
for the first system can be obtained by translating the
6014:
If the equation system is expressed in the matrix form
3330:
Two linear systems using the same set of variables are
3018:
if it has no solution, and otherwise, it is said to be
2502:
For three variables, each linear equation determines a
5917:
5850:
5763:
5693:
5606:
5536:
4492:. There are three types of elementary row operations:
4212:
4138:
2248:
2175:
1985:
1773:
1700:
1617:
1540:
834:{\displaystyle 4\left(3-{\frac {3}{2}}y\right)+9y=15.}
266:
is a system of three equations in the three variables
7770:
7688:
7674:
7511:
7124:
7086:
is sufficiently small, the algorithm is said to have
7049:
7012:
6899:
6859:
6833:
6813:
6793:
6689:
6667:
6645:
6490:
6457:
6430:
6379:
6346:
6316:
6290:
6210:
6180:
6122:
6076:
6020:
5520:
5322:
4539:
4384:
4291:
4191:
4100:
4053:
3919:
3771:
3645:
3503:
3435:
3366:
3135:
3042:
2823:
2742:
2374:
1973:
1906:
1524:
1423:
1361:
1302:
979:
916:
896:
870:
850:
783:
724:
701:
681:
567:
527:
492:
426:
and solutions of the equations are constrained to be
295:
135:
8351:
8200:
7839:Geometrically, this says that the solution set for
7761:, then the entire solution set can be described as
6166:, all solutions (if any exist) are given using the
4182:, equating the RHS of the equations. We now have:
890:, and substituting this back into the equation for
371:for finding the solutions are an important part of
8265:
8239:
8238:Beauregard, Raymond A.; Fraleigh, John B. (1973),
8237:
8130:
8093:
8066:
8054:
7828:
7733:
7530:
7491:
7078:
7035:
6995:
6882:
6845:
6819:
6799:
6697:
6675:
6653:
6628:
6473:
6443:
6411:
6365:
6332:
6298:
6273:
6193:
6138:
6105:
6039:
5987:
5500:
5260:
4473:
4342:
4254:
4167:
4083:
4032:
3862:
3736:, and then computing the corresponding values for
3705:
3625:
3464:
3413:
3259:
3100:
2956:
2800:
2427:such that each of the equations is satisfied. The
2419:
2309:
1925:
1831:
1468:
1409:
1347:
1285:
935:
902:
882:
856:
833:
755:
707:
687:
664:
539:
507:
352:
255:
7095:quantum algorithm for linear systems of equations
6705:give an infinitude of solutions of the equation.
9054:
8533:Elementary Linear Algebra (Applications Version)
6481:and the general solution equation simplifies to
6412:{\displaystyle AA^{+}\mathbf {b} =\mathbf {b} .}
3747:Each free variable gives the solution space one
8264:Burden, Richard L.; Faires, J. Douglas (1993),
6106:{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} }
4178:Since the LHS of both of these equations equal
7748:is any specific solution to the linear system
4515:: Add to one row a scalar multiple of another.
2814:For a more complicated example, the equations
8626:
8308:Golub, Gene H.; Van Loan, Charles F. (1996),
8307:
8013:
2579:of the solution set is, in general, equal to
2517:variables, each linear equation determines a
8263:
8009:
6759:matrix can be solved twice as fast with the
4269:= 2 into the second or third equation gives
3910:For example, consider the following system:
3494:For example, consider the following system:
553:
8560:
7578:
3888:
2604:
2466:
1842:This allows all the language and theory of
1410:{\displaystyle a_{11},a_{12},\dots ,a_{mn}}
8633:
8619:
8490:Matrix Analysis and Applied Linear Algebra
7884:This reasoning only applies if the system
7531:{\displaystyle A\mathbf {x} =\mathbf {0} }
7477:
7473:
7443:
7433:
7403:
7399:
7362:
7337:
7333:
7303:
7293:
7266:
7262:
7227:
7223:
7193:
7183:
7156:
7152:
6366:{\displaystyle A\mathbf {x} =\mathbf {b} }
6040:{\displaystyle A\mathbf {x} =\mathbf {b} }
5834:
5833:
5832:
5831:
5677:
5676:
5675:
5674:
5490:
5484:
5473:
5467:
5456:
5450:
5432:
5426:
5415:
5409:
5398:
5392:
5374:
5368:
5357:
5351:
5340:
5334:
5313:. For example, the solution to the system
4372:), the linear system is represented as an
3819:
3818:
3817:
3816:
3810:
3809:
3808:
3807:
3679:
3678:
3677:
3676:
3670:
3669:
3668:
3667:
3611:
3607:
3594:
3590:
3577:
3573:
3552:
3548:
3535:
3531:
3518:
3514:
3398:
3382:
3354:
3338:
3245:
3241:
3228:
3224:
3203:
3199:
3189:
3185:
3164:
3160:
3150:
3146:
3076:
3075:
3074:
3073:
3067:
3066:
3065:
3064:
2942:
2938:
2925:
2921:
2900:
2896:
2883:
2879:
2855:
2851:
2838:
2834:
2776:
2775:
2774:
2773:
2767:
2766:
2765:
2764:
2532:, which may have any dimension lower than
1926:{\displaystyle A\mathbf {x} =\mathbf {b} }
644:
640:
627:
623:
602:
598:
585:
581:
483:The system of one equation in one unknown
360:since it makes all three equations valid.
8574:
8365:
7877:for the homogeneous system by the vector
6333:{\displaystyle \mathbf {w} =\mathbf {0} }
5978:
5911:
5908:
5844:
5824:
5757:
5754:
5687:
5667:
5600:
5597:
5530:
3455:
3445:
3318:reduced to a system that has a number of
3293:Putting it another way, according to the
2665:and has no more equations than unknowns.
363:Linear systems are a fundamental part of
102:. The intersection point is the solution.
82:Learn how and when to remove this message
8423:
8405:
8158:
8142:
8111:
7978:(NAG Library versions of LAPACK solvers)
6154:. More generally, regardless of whether
2981:
2689:
2621:
2614:
2607:
2543:
2431:of all possible solutions is called the
2420:{\displaystyle x_{1},x_{2},\dots ,x_{n}}
2330:
1479:Often the coefficients and unknowns are
1469:{\displaystyle b_{1},b_{2},\dots ,b_{m}}
1417:are the coefficients of the system, and
1348:{\displaystyle x_{1},x_{2},\dots ,x_{n}}
93:
45:This article includes a list of general
7115:if all of the constant terms are zero:
6775:), which appear often in applications.
3115:. The graphs of these equations on the
1896:The vector equation is equivalent to a
14:
9055:
9024:Comparison of linear algebra libraries
8548:
8544:(7th ed.). Pearson Prentice Hall.
8329:
8289:
8099:
8017:
7100:
3751:, the number of which is equal to the
2471:For a system involving two variables (
473:
18:Homogeneous system of linear equations
8614:
8530:
8514:Linear Algebra: A Modern Introduction
8511:
8486:
8448:
8215:
8005:
7079:{\displaystyle {\mathbf {x}}^{(k+1)}}
6807:is split into its diagonal component
6162:or not and regardless of the rank of
3491:on the values of the free variables.
2978:Consistent and inconsistent equations
2678:The equations of a linear system are
2479:), each linear equation determines a
422:Very often, and in this article, the
407:), a helpful technique when making a
8539:
8535:(9th ed.). Wiley International.
8487:Meyer, Carl D. (February 15, 2001).
8332:Introduction to Mathematical Physics
7006:When the difference between guesses
6715:Numerical solution of linear systems
756:{\displaystyle x=3-{\frac {3}{2}}y.}
31:
8554:Linear Algebra and Its Applications
8470:Linear Algebra and Its Applications
8467:
8292:Matrices and Linear Transformations
8201:Harrow, Hassidim & Lloyd (2009)
7036:{\displaystyle {\mathbf {x}}^{(k)}}
6883:{\displaystyle {\mathbf {x}}^{(0)}}
4484:This matrix is then modified using
3425:the solution may be described as a
2539:
2335:The solution set for the equations
24:
8640:
8442:
7675:Relation to nonhomogeneous systems
6009:
3414:{\displaystyle (x=3,\;y=-2,\;z=6)}
1891:
1502:
478:
353:{\displaystyle (x,y,z)=(1,-2,-2),}
51:it lacks sufficient corresponding
25:
9084:
8592:
8468:Lay, David C. (August 22, 2005).
8453:(2nd ed.). Springer-Verlag.
8159:Hartnett, Kevin (March 8, 2021).
7651:is also a solution to the system.
7629:is also a solution to the system.
7107:Homogeneous differential equation
4499:: Swap the positions of two rows.
4343:{\displaystyle (x,y,z)=(-15,8,2)}
118:) is a collection of two or more
9037:
9036:
9014:Basic Linear Algebra Subprograms
8772:
8598:
8542:Linear Algebra With Applications
8472:(3rd ed.). Addison Wesley.
8410:, Indianapolis, Indiana: Wiley,
8131:Beauregard & Fraleigh (1973)
8067:Beauregard & Fraleigh (1973)
8055:Beauregard & Fraleigh (1973)
7814:
7806:
7793:
7785:
7777:
7724:
7716:
7701:
7693:
7524:
7516:
7111:A system of linear equations is
7053:
7016:
6973:
6947:
6903:
6863:
6708:
6691:
6669:
6647:
6622:
6601:
6574:
6553:
6513:
6492:
6402:
6394:
6359:
6351:
6326:
6318:
6292:
6267:
6230:
6212:
6099:
6078:
6063:columns) and has full rank (all
6033:
6025:
5295:
4353:
3305:is greater than the rank of the
2661:
2622:
2615:
2608:
2358:is the single point (2, 3).
2236:
2163:
1919:
1911:
936:{\displaystyle x={\frac {3}{2}}}
36:
8912:Seven-dimensional cross product
8209:
8180:
8152:
7711:
7705:
6827:and its non-diagonal component
4474:{\displaystyle \left{\text{.}}}
4281:into the first equation yields
4043:Solving the first equation for
3322:that is at most equal to + 1.
2673:
2593:is the number of variables and
2326:
2234:
2161:
950:
375:, and play a prominent role in
8426:Introduction to Linear Algebra
8384:10.1103/PhysRevLett.103.150502
8314:Johns Hopkins University Press
8072:
8023:
7799: is any solution to
7071:
7059:
7028:
7022:
6990:
6985:
6979:
6967:
6955:
6942:
6921:
6909:
6875:
6869:
4505:: Multiply a row by a nonzero
4337:
4316:
4310:
4292:
4236:
3459:
3436:
3408:
3367:
3351:a system of linear equations.
3325:
2971:
2650:intersect at a single point).
397:system of non-linear equations
344:
320:
314:
296:
13:
1:
8516:(2nd ed.). Brooks/Cole.
8080:"Systems of Linear Equations"
7993:
2668:
8754:Eigenvalues and eigenvectors
7667:of the corresponding matrix
6698:{\displaystyle \mathbf {w} }
6676:{\displaystyle \mathbf {w} }
6654:{\displaystyle \mathbf {w} }
6639:as previously stated, where
6424:is square and of full rank,
6299:{\displaystyle \mathbf {w} }
3720:is the free variable, while
3465:{\displaystyle (3,\,-2,\,6)}
2636:
2633:
2630:
2597:is the number of equations.
7:
8449:Axler, Sheldon Jay (1997).
8428:(2nd ed.), CRC Press,
8312:(3rd ed.), Baltimore:
8290:Cullen, Charles G. (1990),
8014:Golub & Van Loan (1996)
7956:Linear equation over a ring
7915:
3033:For example, the equations
2733:For example, the equations
456:Linear equation over a ring
10:
9089:
8605:System of linear equations
8408:Linear Algebra for Dummies
8406:Sterling, Mary J. (2009),
8274:Prindle, Weber and Schmidt
8220:(5th ed.), New York:
8010:Burden & Faires (1993)
7923:Arrangement of hyperplanes
7104:
6712:
5299:
4357:
3877:is the free variable, and
3472:for the previous example.
2975:
774:into the bottom equation:
464:Integer linear programming
112:system of linear equations
9032:
8994:
8950:
8887:
8839:
8781:
8770:
8666:
8648:
8451:Linear Algebra Done Right
8218:Elementary Linear Algebra
4486:elementary row operations
4084:{\displaystyle x=5+2z-3y}
2446:infinitely many solutions
554:Simple nontrivial example
9073:Numerical linear algebra
8540:Leon, Steven J. (2006).
8424:Whitelaw, T. A. (1991),
8330:Harper, Charlie (1976),
8272:(5th ed.), Boston:
8248:Houghton Mifflin Company
7856:of the solution set for
7579:Homogeneous solution set
7559:is a column vector with
4528:Gauss–Jordan elimination
4490:reduced row echelon form
3889:Elimination of variables
2467:Geometric interpretation
1961:is a column vector with
1876:) cannot be larger than
1476:are the constant terms.
403:by a linear system (see
373:numerical linear algebra
8354:Physical Review Letters
7982:Rybicki Press algorithm
4273:= 8, and the values of
3355:Describing the solution
3339:Solving a linear system
2729:are linearly dependent.
2575:In the first case, the
2508:three-dimensional space
66:more precise citations.
8739:Row and column vectors
8531:Anton, Howard (2005).
8216:Anton, Howard (1987),
7987:Simultaneous equations
7830:
7735:
7532:
7493:
7080:
7037:
6997:
6884:
6847:
6821:
6801:
6761:Cholesky decomposition
6741:but different vectors
6699:
6677:
6655:
6630:
6475:
6474:{\displaystyle A^{-1}}
6445:
6413:
6373:— that is, that
6367:
6334:
6300:
6275:
6195:
6140:
6139:{\displaystyle A^{-1}}
6107:
6041:
6000:matrix of coefficients
5989:
5502:
5262:
4475:
4344:
4256:
4169:
4085:
4034:
3864:
3707:
3627:
3466:
3415:
3313:free parameters where
3295:Rouché–Capelli theorem
3261:
3102:
3011:
2958:
2802:
2730:
2559:underdetermined system
2549:
2421:
2359:
2311:
1927:
1833:
1470:
1411:
1349:
1287:
959:linear equations with
937:
904:
884:
858:
835:
757:
709:
689:
666:
541:
509:
354:
257:
103:
8744:Row and column spaces
8689:Scalar multiplication
8512:Poole, David (2006).
8031:"System of Equations"
7976:NAG Numerical Library
7907:linear transformation
7831:
7736:
7533:
7494:
7081:
7038:
6998:
6885:
6848:
6822:
6802:
6767:is a fast method for
6713:Further information:
6700:
6678:
6656:
6631:
6476:
6446:
6444:{\displaystyle A^{+}}
6414:
6368:
6335:
6301:
6276:
6196:
6194:{\displaystyle A^{+}}
6168:Moore–Penrose inverse
6141:
6108:
6042:
5990:
5503:
5263:
4476:
4345:
4257:
4170:
4086:
4035:
3865:
3708:
3628:
3467:
3416:
3320:independent equations
3262:
3119:-plane are a pair of
3103:
2985:
2959:
2803:
2693:
2569:overdetermined system
2547:
2422:
2334:
2312:
1928:
1900:equation of the form
1834:
1471:
1412:
1350:
1288:
938:
905:
885:
859:
836:
758:
710:
690:
667:
542:
510:
355:
258:
97:
8879:Gram–Schmidt process
8831:Gaussian elimination
8607:at Wikimedia Commons
7966:Matrix decomposition
7961:Linear least squares
7928:Iterative refinement
7869:. Specifically, the
7768:
7686:
7509:
7122:
7047:
7010:
6897:
6857:
6831:
6811:
6791:
6687:
6665:
6643:
6488:
6455:
6428:
6377:
6344:
6314:
6288:
6208:
6178:
6120:
6074:
6018:
5518:
5320:
4537:
4524:Gaussian elimination
4382:
4370:Gaussian elimination
4360:Gaussian elimination
4289:
4189:
4098:
4051:
3917:
3769:
3643:
3501:
3433:
3364:
3133:
3040:
2821:
2740:
2372:
1971:
1904:
1868:linearly independent
1848:(or more generally,
1522:
1421:
1359:
1300:
977:
955:A general system of
914:
894:
868:
848:
781:
770:this expression for
722:
699:
679:
565:
540:{\displaystyle x=2.}
525:
508:{\displaystyle 2x=4}
490:
440:algebraic structures
293:
133:
9009:Numerical stability
8889:Multilinear algebra
8864:Inner product space
8714:Linear independence
8376:2009PhRvL.103o0502H
8310:Matrix Computations
7593:non-singular matrix
7101:Homogeneous systems
6853:. An initial guess
6846:{\displaystyle L+U}
6787:, where the matrix
3014:A linear system is
2685:linear independence
1497:algebraic structure
883:{\displaystyle y=1}
474:Elementary examples
413:computer simulation
122:involving the same
8719:Linear combination
8268:Numerical Analysis
7826:
7731:
7528:
7489:
7487:
7076:
7033:
6993:
6880:
6843:
6817:
6797:
6765:Levinson recursion
6695:
6673:
6651:
6626:
6471:
6441:
6409:
6363:
6330:
6296:
6271:
6191:
6136:
6103:
6037:
5985:
5972:
5902:
5818:
5748:
5661:
5591:
5498:
5496:
5258:
5256:
5245:
5161:
5080:
4996:
4902:
4809:
4713:
4619:
4471:
4460:
4340:
4252:
4250:
4221:
4165:
4160:
4147:
4081:
4030:
4025:
3860:
3703:
3623:
3621:
3462:
3423:alphabetical order
3411:
3343:There are several
3307:coefficient matrix
3257:
3255:
3098:
3012:
2954:
2952:
2798:
2731:
2656:linearly dependent
2550:
2526:-dimensional space
2417:
2360:
2307:
2298:
2225:
2152:
1923:
1829:
1823:
1759:
1670:
1593:
1513:linear combination
1466:
1407:
1355:are the unknowns,
1345:
1283:
1278:
970:can be written as
945:elementary algebra
933:
900:
880:
854:
831:
753:
705:
685:
662:
660:
537:
518:has the solution
505:
409:mathematical model
350:
253:
248:
104:
9050:
9049:
8917:Geometric algebra
8874:Kronecker product
8709:Linear projection
8694:Vector projection
8603:Media related to
8507:on March 1, 2001.
8500:978-0-89871-454-8
8479:978-0-321-28713-7
8417:978-0-470-43090-3
8301:978-0-486-66328-9
8087:math.berkeley.edu
8069:, pp. 65–66.
7800:
7744:Specifically, if
7709:
7365:
7090:on the solution.
6820:{\displaystyle D}
6800:{\displaystyle A}
6781:iterative methods
6769:Toeplitz matrices
6757:positive definite
5980:
5826:
5669:
4488:until it reaches
4469:
4220:
4146:
3858:
3853:
3837:
3814:
3805:
3789:
3749:degree of freedom
3728:are dependent on
3701:
3674:
3071:
3010:are inconsistent.
2771:
2641:
2640:
2451:The system has a
931:
903:{\displaystyle x}
857:{\displaystyle y}
806:
745:
708:{\displaystyle y}
688:{\displaystyle x}
468:Tropical geometry
229:
92:
91:
84:
16:(Redirected from
9080:
9040:
9039:
8922:Exterior algebra
8859:Hadamard product
8776:
8764:Linear equations
8635:
8628:
8621:
8612:
8611:
8602:
8588:
8578:
8557:
8545:
8536:
8527:
8508:
8503:. Archived from
8483:
8464:
8438:
8420:
8402:
8369:
8348:
8326:
8304:
8286:
8271:
8260:
8245:
8234:
8204:
8198:
8192:
8191:
8184:
8178:
8177:
8175:
8173:
8156:
8150:
8140:
8134:
8128:
8119:
8109:
8103:
8097:
8091:
8090:
8084:
8076:
8070:
8064:
8058:
8052:
8046:
8045:
8043:
8041:
8027:
8021:
8003:
7971:Matrix splitting
7896:
7868:
7851:
7835:
7833:
7832:
7827:
7822:
7818:
7817:
7809:
7801:
7798:
7796:
7788:
7780:
7760:
7740:
7738:
7737:
7732:
7727:
7719:
7710:
7707:
7704:
7696:
7628:
7602:
7554:
7537:
7535:
7534:
7529:
7527:
7519:
7498:
7496:
7495:
7490:
7488:
7480:
7479:
7471:
7469:
7468:
7459:
7458:
7445:
7431:
7429:
7428:
7419:
7418:
7405:
7397:
7395:
7394:
7385:
7384:
7369:
7368:
7367:
7363:
7357:
7356:
7355:
7354:
7353:
7352:
7351:
7350:
7349:
7348:
7340:
7339:
7331:
7329:
7328:
7319:
7318:
7305:
7291:
7289:
7288:
7279:
7278:
7268:
7260:
7258:
7257:
7248:
7247:
7230:
7229:
7221:
7219:
7218:
7209:
7208:
7195:
7181:
7179:
7178:
7169:
7168:
7158:
7150:
7148:
7147:
7138:
7137:
7093:There is also a
7085:
7083:
7082:
7077:
7075:
7074:
7057:
7056:
7042:
7040:
7039:
7034:
7032:
7031:
7020:
7019:
7002:
7000:
6999:
6994:
6989:
6988:
6977:
6976:
6951:
6950:
6941:
6940:
6925:
6924:
6907:
6906:
6889:
6887:
6886:
6881:
6879:
6878:
6867:
6866:
6852:
6850:
6849:
6844:
6826:
6824:
6823:
6818:
6806:
6804:
6803:
6798:
6731:LU decomposition
6704:
6702:
6701:
6696:
6694:
6682:
6680:
6679:
6674:
6672:
6660:
6658:
6657:
6652:
6650:
6635:
6633:
6632:
6627:
6625:
6620:
6619:
6604:
6599:
6595:
6577:
6572:
6571:
6556:
6551:
6547:
6543:
6542:
6516:
6511:
6510:
6495:
6480:
6478:
6477:
6472:
6470:
6469:
6450:
6448:
6447:
6442:
6440:
6439:
6418:
6416:
6415:
6410:
6405:
6397:
6392:
6391:
6372:
6370:
6369:
6364:
6362:
6354:
6339:
6337:
6336:
6331:
6329:
6321:
6305:
6303:
6302:
6297:
6295:
6280:
6278:
6277:
6272:
6270:
6265:
6261:
6257:
6256:
6233:
6228:
6227:
6215:
6200:
6198:
6197:
6192:
6190:
6189:
6145:
6143:
6142:
6137:
6135:
6134:
6112:
6110:
6109:
6104:
6102:
6097:
6096:
6081:
6046:
6044:
6043:
6038:
6036:
6028:
5994:
5992:
5991:
5986:
5981:
5979:
5977:
5976:
5909:
5907:
5906:
5842:
5827:
5825:
5823:
5822:
5755:
5753:
5752:
5685:
5670:
5668:
5666:
5665:
5598:
5596:
5595:
5528:
5507:
5505:
5504:
5499:
5497:
5291:
5284:
5277:
5267:
5265:
5264:
5259:
5257:
5250:
5246:
5166:
5162:
5085:
5081:
5001:
4997:
4911:
4907:
4903:
4814:
4810:
4718:
4714:
4624:
4620:
4480:
4478:
4477:
4472:
4470:
4467:
4465:
4461:
4374:augmented matrix
4349:
4347:
4346:
4341:
4261:
4259:
4258:
4253:
4251:
4222:
4213:
4174:
4172:
4171:
4166:
4164:
4163:
4148:
4139:
4090:
4088:
4087:
4082:
4039:
4037:
4036:
4031:
4029:
4028:
3869:
3867:
3866:
3861:
3859:
3856:
3854:
3846:
3838:
3830:
3815:
3812:
3806:
3798:
3790:
3782:
3712:
3710:
3709:
3704:
3702:
3699:
3675:
3672:
3632:
3630:
3629:
3624:
3622:
3619:
3613:
3605:
3596:
3588:
3579:
3571:
3560:
3554:
3546:
3537:
3529:
3520:
3512:
3471:
3469:
3468:
3463:
3429:of values, like
3420:
3418:
3417:
3412:
3303:augmented matrix
3285:
3281:
3266:
3264:
3263:
3258:
3256:
3253:
3247:
3239:
3230:
3222:
3211:
3205:
3197:
3191:
3183:
3172:
3166:
3158:
3152:
3144:
3114:
3107:
3105:
3104:
3099:
3072:
3069:
3029:
3009:
2997:
2963:
2961:
2960:
2955:
2953:
2950:
2944:
2936:
2927:
2919:
2908:
2902:
2894:
2885:
2877:
2866:
2857:
2849:
2840:
2832:
2807:
2805:
2804:
2799:
2772:
2769:
2728:
2716:
2704:
2637:Three equations
2626:
2619:
2612:
2605:
2588:
2540:General behavior
2426:
2424:
2423:
2418:
2416:
2415:
2397:
2396:
2384:
2383:
2357:
2345:
2316:
2314:
2313:
2308:
2303:
2302:
2295:
2294:
2274:
2273:
2260:
2259:
2239:
2230:
2229:
2222:
2221:
2201:
2200:
2187:
2186:
2166:
2157:
2156:
2149:
2148:
2129:
2128:
2114:
2113:
2075:
2074:
2055:
2054:
2043:
2042:
2029:
2028:
2009:
2008:
1997:
1996:
1932:
1930:
1929:
1924:
1922:
1914:
1838:
1836:
1835:
1830:
1828:
1827:
1820:
1819:
1799:
1798:
1785:
1784:
1764:
1763:
1756:
1755:
1732:
1731:
1715:
1714:
1694:
1693:
1675:
1674:
1667:
1666:
1643:
1642:
1629:
1628:
1611:
1610:
1598:
1597:
1590:
1589:
1566:
1565:
1552:
1551:
1534:
1533:
1493:rational numbers
1475:
1473:
1472:
1467:
1465:
1464:
1446:
1445:
1433:
1432:
1416:
1414:
1413:
1408:
1406:
1405:
1384:
1383:
1371:
1370:
1354:
1352:
1351:
1346:
1344:
1343:
1325:
1324:
1312:
1311:
1292:
1290:
1289:
1284:
1282:
1281:
1272:
1271:
1259:
1258:
1249:
1248:
1227:
1226:
1217:
1216:
1201:
1200:
1191:
1190:
1167:
1166:
1154:
1153:
1144:
1143:
1122:
1121:
1112:
1111:
1099:
1098:
1089:
1088:
1075:
1074:
1062:
1061:
1052:
1051:
1030:
1029:
1020:
1019:
1007:
1006:
997:
996:
942:
940:
939:
934:
932:
924:
909:
907:
906:
901:
889:
887:
886:
881:
864:. Solving gives
863:
861:
860:
855:
840:
838:
837:
832:
815:
811:
807:
799:
762:
760:
759:
754:
746:
738:
714:
712:
711:
706:
694:
692:
691:
686:
671:
669:
668:
663:
661:
646:
638:
629:
621:
610:
604:
596:
587:
579:
546:
544:
543:
538:
514:
512:
511:
506:
415:of a relatively
389:computer science
359:
357:
356:
351:
279:
262:
260:
259:
254:
252:
251:
230:
222:
120:linear equations
87:
80:
76:
73:
67:
62:this article by
53:inline citations
40:
39:
32:
21:
9088:
9087:
9083:
9082:
9081:
9079:
9078:
9077:
9053:
9052:
9051:
9046:
9028:
8990:
8946:
8883:
8835:
8777:
8768:
8734:Change of basis
8724:Multilinear map
8662:
8644:
8639:
8595:
8585:10.1145/3615679
8550:Strang, Gilbert
8524:
8501:
8480:
8461:
8445:
8443:Further reading
8436:
8418:
8346:
8324:
8302:
8284:
8258:
8232:
8212:
8207:
8199:
8195:
8188:"Jacobi Method"
8186:
8185:
8181:
8171:
8169:
8166:Quanta Magazine
8157:
8153:
8143:Sterling (2009)
8141:
8137:
8129:
8122:
8112:Whitelaw (1991)
8110:
8106:
8098:
8094:
8082:
8078:
8077:
8073:
8065:
8061:
8053:
8049:
8039:
8037:
8029:
8028:
8024:
8012:, p. 324;
8004:
8000:
7996:
7991:
7918:
7885:
7875:linear subspace
7857:
7840:
7813:
7805:
7797:
7792:
7784:
7776:
7775:
7771:
7769:
7766:
7765:
7749:
7723:
7715:
7706:
7700:
7692:
7687:
7684:
7683:
7677:
7657:linear subspace
7620:
7596:
7581:
7546:
7523:
7515:
7510:
7507:
7506:
7486:
7485:
7478:
7470:
7464:
7460:
7451:
7447:
7444:
7430:
7424:
7420:
7411:
7407:
7404:
7396:
7390:
7386:
7377:
7373:
7370:
7366:
7346:
7345:
7338:
7330:
7324:
7320:
7311:
7307:
7304:
7290:
7284:
7280:
7274:
7270:
7267:
7259:
7253:
7249:
7243:
7239:
7236:
7235:
7228:
7220:
7214:
7210:
7201:
7197:
7194:
7180:
7174:
7170:
7164:
7160:
7157:
7149:
7143:
7139:
7133:
7129:
7125:
7123:
7120:
7119:
7109:
7103:
7058:
7052:
7051:
7050:
7048:
7045:
7044:
7021:
7015:
7014:
7013:
7011:
7008:
7007:
6978:
6972:
6971:
6970:
6946:
6945:
6933:
6929:
6908:
6902:
6901:
6900:
6898:
6895:
6894:
6868:
6862:
6861:
6860:
6858:
6855:
6854:
6832:
6829:
6828:
6812:
6809:
6808:
6792:
6789:
6788:
6773:sparse matrices
6717:
6711:
6690:
6688:
6685:
6684:
6668:
6666:
6663:
6662:
6646:
6644:
6641:
6640:
6621:
6612:
6608:
6600:
6585:
6581:
6573:
6564:
6560:
6552:
6535:
6531:
6524:
6520:
6512:
6503:
6499:
6491:
6489:
6486:
6485:
6462:
6458:
6456:
6453:
6452:
6435:
6431:
6429:
6426:
6425:
6401:
6393:
6387:
6383:
6378:
6375:
6374:
6358:
6350:
6345:
6342:
6341:
6325:
6317:
6315:
6312:
6311:
6291:
6289:
6286:
6285:
6266:
6252:
6248:
6241:
6237:
6229:
6223:
6219:
6211:
6209:
6206:
6205:
6185:
6181:
6179:
6176:
6175:
6127:
6123:
6121:
6118:
6117:
6098:
6089:
6085:
6077:
6075:
6072:
6071:
6051:is square (has
6032:
6024:
6019:
6016:
6015:
6012:
6010:Matrix solution
5971:
5970:
5965:
5960:
5954:
5953:
5948:
5943:
5937:
5936:
5928:
5923:
5913:
5912:
5910:
5901:
5900:
5895:
5890:
5884:
5883:
5878:
5873:
5867:
5866:
5861:
5856:
5846:
5845:
5843:
5841:
5817:
5816:
5811:
5806:
5800:
5799:
5794:
5789:
5783:
5782:
5774:
5769:
5759:
5758:
5756:
5747:
5746:
5741:
5736:
5730:
5729:
5724:
5719:
5713:
5712:
5704:
5699:
5689:
5688:
5686:
5684:
5660:
5659:
5654:
5649:
5643:
5642:
5637:
5632:
5626:
5625:
5617:
5612:
5602:
5601:
5599:
5590:
5589:
5584:
5579:
5573:
5572:
5567:
5562:
5556:
5555:
5547:
5542:
5532:
5531:
5529:
5527:
5519:
5516:
5515:
5495:
5494:
5488:
5480:
5471:
5463:
5454:
5446:
5437:
5436:
5430:
5422:
5413:
5405:
5396:
5388:
5379:
5378:
5372:
5364:
5355:
5347:
5338:
5330:
5323:
5321:
5318:
5317:
5304:
5298:
5286:
5279:
5272:
5255:
5254:
5244:
5243:
5238:
5233:
5228:
5222:
5221:
5216:
5211:
5206:
5200:
5199:
5191:
5186:
5181:
5174:
5170:
5160:
5159:
5154:
5149:
5144:
5138:
5137:
5132:
5127:
5122:
5116:
5115:
5110:
5105:
5100:
5093:
5089:
5079:
5078:
5073:
5068:
5063:
5057:
5056:
5051:
5046:
5041:
5035:
5034:
5029:
5021:
5016:
5009:
5005:
4995:
4994:
4989:
4984:
4979:
4973:
4972:
4967:
4959:
4954:
4948:
4947:
4942:
4934:
4929:
4922:
4918:
4909:
4908:
4901:
4900:
4892:
4887:
4879:
4873:
4872:
4867:
4859:
4854:
4848:
4847:
4842:
4834:
4829:
4822:
4818:
4808:
4807:
4799:
4794:
4786:
4780:
4779:
4771:
4766:
4758:
4752:
4751:
4746:
4738:
4733:
4726:
4722:
4712:
4711:
4706:
4701:
4696:
4690:
4689:
4681:
4676:
4668:
4662:
4661:
4656:
4648:
4643:
4636:
4632:
4625:
4618:
4617:
4612:
4607:
4602:
4596:
4595:
4590:
4585:
4580:
4574:
4573:
4568:
4560:
4555:
4548:
4544:
4540:
4538:
4535:
4534:
4466:
4459:
4458:
4453:
4448:
4443:
4437:
4436:
4431:
4426:
4421:
4415:
4414:
4409:
4401:
4396:
4389:
4385:
4383:
4380:
4379:
4368:(also known as
4362:
4356:
4290:
4287:
4286:
4249:
4248:
4233:
4232:
4211:
4192:
4190:
4187:
4186:
4159:
4158:
4137:
4128:
4127:
4102:
4101:
4099:
4096:
4095:
4052:
4049:
4048:
4024:
4023:
3990:
3989:
3956:
3955:
3921:
3920:
3918:
3915:
3914:
3891:
3885:are dependent.
3855:
3845:
3829:
3811:
3797:
3781:
3770:
3767:
3766:
3698:
3671:
3644:
3641:
3640:
3620:
3618:
3612:
3604:
3595:
3587:
3578:
3570:
3561:
3559:
3553:
3545:
3536:
3528:
3519:
3511:
3504:
3502:
3499:
3498:
3434:
3431:
3430:
3365:
3362:
3361:
3357:
3341:
3328:
3283:
3271:
3254:
3252:
3246:
3238:
3229:
3221:
3212:
3210:
3204:
3196:
3190:
3182:
3173:
3171:
3165:
3157:
3151:
3143:
3136:
3134:
3131:
3130:
3112:
3068:
3041:
3038:
3037:
3027:
2999:
2987:
2980:
2974:
2951:
2949:
2943:
2935:
2926:
2918:
2909:
2907:
2901:
2893:
2884:
2876:
2867:
2865:
2856:
2848:
2839:
2831:
2824:
2822:
2819:
2818:
2768:
2741:
2738:
2737:
2718:
2706:
2695:
2676:
2671:
2580:
2542:
2469:
2458:The system has
2453:unique solution
2444:The system has
2411:
2407:
2392:
2388:
2379:
2375:
2373:
2370:
2369:
2347:
2336:
2329:
2323:of the matrix.
2297:
2296:
2290:
2286:
2283:
2282:
2276:
2275:
2269:
2265:
2262:
2261:
2255:
2251:
2244:
2243:
2235:
2224:
2223:
2217:
2213:
2210:
2209:
2203:
2202:
2196:
2192:
2189:
2188:
2182:
2178:
2171:
2170:
2162:
2151:
2150:
2141:
2137:
2135:
2130:
2121:
2117:
2115:
2106:
2102:
2099:
2098:
2093:
2088:
2083:
2077:
2076:
2067:
2063:
2061:
2056:
2050:
2046:
2044:
2038:
2034:
2031:
2030:
2021:
2017:
2015:
2010:
2004:
2000:
1998:
1992:
1988:
1981:
1980:
1972:
1969:
1968:
1918:
1910:
1905:
1902:
1901:
1894:
1892:Matrix equation
1822:
1821:
1815:
1811:
1808:
1807:
1801:
1800:
1794:
1790:
1787:
1786:
1780:
1776:
1769:
1768:
1758:
1757:
1748:
1744:
1741:
1740:
1734:
1733:
1724:
1720:
1717:
1716:
1707:
1703:
1696:
1695:
1689:
1685:
1669:
1668:
1659:
1655:
1652:
1651:
1645:
1644:
1638:
1634:
1631:
1630:
1624:
1620:
1613:
1612:
1606:
1602:
1592:
1591:
1582:
1578:
1575:
1574:
1568:
1567:
1561:
1557:
1554:
1553:
1547:
1543:
1536:
1535:
1529:
1525:
1523:
1520:
1519:
1505:
1503:Vector equation
1485:complex numbers
1460:
1456:
1441:
1437:
1428:
1424:
1422:
1419:
1418:
1398:
1394:
1379:
1375:
1366:
1362:
1360:
1357:
1356:
1339:
1335:
1320:
1316:
1307:
1303:
1301:
1298:
1297:
1277:
1276:
1267:
1263:
1254:
1250:
1241:
1237:
1222:
1218:
1209:
1205:
1196:
1192:
1183:
1179:
1176:
1175:
1169:
1168:
1162:
1158:
1149:
1145:
1136:
1132:
1117:
1113:
1107:
1103:
1094:
1090:
1084:
1080:
1077:
1076:
1070:
1066:
1057:
1053:
1044:
1040:
1025:
1021:
1015:
1011:
1002:
998:
992:
988:
981:
980:
978:
975:
974:
953:
923:
915:
912:
911:
895:
892:
891:
869:
866:
865:
849:
846:
845:
798:
791:
787:
782:
779:
778:
737:
723:
720:
719:
700:
697:
696:
680:
677:
676:
659:
658:
651:
645:
637:
628:
620:
611:
609:
603:
595:
586:
578:
568:
566:
563:
562:
556:
526:
523:
522:
491:
488:
487:
481:
479:Trivial example
476:
444:integral domain
432:complex numbers
294:
291:
290:
267:
247:
246:
221:
209:
208:
172:
171:
137:
136:
134:
131:
130:
126:. For example,
88:
77:
71:
68:
58:Please help to
57:
41:
37:
28:
23:
22:
15:
12:
11:
5:
9086:
9076:
9075:
9070:
9068:Linear algebra
9065:
9048:
9047:
9045:
9044:
9033:
9030:
9029:
9027:
9026:
9021:
9016:
9011:
9006:
9004:Floating-point
9000:
8998:
8992:
8991:
8989:
8988:
8986:Tensor product
8983:
8978:
8973:
8971:Function space
8968:
8963:
8957:
8955:
8948:
8947:
8945:
8944:
8939:
8934:
8929:
8924:
8919:
8914:
8909:
8907:Triple product
8904:
8899:
8893:
8891:
8885:
8884:
8882:
8881:
8876:
8871:
8866:
8861:
8856:
8851:
8845:
8843:
8837:
8836:
8834:
8833:
8828:
8823:
8821:Transformation
8818:
8813:
8811:Multiplication
8808:
8803:
8798:
8793:
8787:
8785:
8779:
8778:
8771:
8769:
8767:
8766:
8761:
8756:
8751:
8746:
8741:
8736:
8731:
8726:
8721:
8716:
8711:
8706:
8701:
8696:
8691:
8686:
8681:
8676:
8670:
8668:
8667:Basic concepts
8664:
8663:
8661:
8660:
8655:
8649:
8646:
8645:
8642:Linear algebra
8638:
8637:
8630:
8623:
8615:
8609:
8608:
8594:
8593:External links
8591:
8590:
8589:
8558:
8546:
8537:
8528:
8522:
8509:
8499:
8484:
8478:
8465:
8459:
8444:
8441:
8440:
8439:
8434:
8421:
8416:
8403:
8360:(15): 150502,
8349:
8344:
8334:, New Jersey:
8327:
8322:
8305:
8300:
8287:
8282:
8261:
8256:
8235:
8230:
8211:
8208:
8206:
8205:
8193:
8179:
8151:
8135:
8120:
8104:
8092:
8071:
8059:
8047:
8022:
8016:, p. 87;
7997:
7995:
7992:
7990:
7989:
7984:
7979:
7973:
7968:
7963:
7958:
7953:
7935:
7930:
7925:
7919:
7917:
7914:
7837:
7836:
7825:
7821:
7816:
7812:
7808:
7804:
7795:
7791:
7787:
7783:
7779:
7774:
7742:
7741:
7730:
7726:
7722:
7718:
7714:
7703:
7699:
7695:
7691:
7676:
7673:
7653:
7652:
7630:
7580:
7577:
7539:
7538:
7526:
7522:
7518:
7514:
7500:
7499:
7484:
7481:
7476:
7472:
7467:
7463:
7457:
7454:
7450:
7446:
7442:
7439:
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7432:
7427:
7423:
7417:
7414:
7410:
7406:
7402:
7398:
7393:
7389:
7383:
7380:
7376:
7372:
7371:
7361:
7358:
7347:
7344:
7341:
7336:
7332:
7327:
7323:
7317:
7314:
7310:
7306:
7302:
7299:
7296:
7292:
7287:
7283:
7277:
7273:
7269:
7265:
7261:
7256:
7252:
7246:
7242:
7238:
7237:
7234:
7231:
7226:
7222:
7217:
7213:
7207:
7204:
7200:
7196:
7192:
7189:
7186:
7182:
7177:
7173:
7167:
7163:
7159:
7155:
7151:
7146:
7142:
7136:
7132:
7128:
7127:
7102:
7099:
7073:
7070:
7067:
7064:
7061:
7055:
7030:
7027:
7024:
7018:
7004:
7003:
6992:
6987:
6984:
6981:
6975:
6969:
6966:
6963:
6960:
6957:
6954:
6949:
6944:
6939:
6936:
6932:
6928:
6923:
6920:
6917:
6914:
6911:
6905:
6877:
6874:
6871:
6865:
6842:
6839:
6836:
6816:
6796:
6748:If the matrix
6733:of the matrix
6710:
6707:
6693:
6671:
6649:
6637:
6636:
6624:
6618:
6615:
6611:
6607:
6603:
6598:
6594:
6591:
6588:
6584:
6580:
6576:
6570:
6567:
6563:
6559:
6555:
6550:
6546:
6541:
6538:
6534:
6530:
6527:
6523:
6519:
6515:
6509:
6506:
6502:
6498:
6494:
6468:
6465:
6461:
6451:simply equals
6438:
6434:
6408:
6404:
6400:
6396:
6390:
6386:
6382:
6361:
6357:
6353:
6349:
6328:
6324:
6320:
6294:
6282:
6281:
6269:
6264:
6260:
6255:
6251:
6247:
6244:
6240:
6236:
6232:
6226:
6222:
6218:
6214:
6201:, as follows:
6188:
6184:
6133:
6130:
6126:
6114:
6113:
6101:
6095:
6092:
6088:
6084:
6080:
6035:
6031:
6027:
6023:
6011:
6008:
5996:
5995:
5984:
5975:
5969:
5966:
5964:
5961:
5959:
5956:
5955:
5952:
5949:
5947:
5944:
5942:
5939:
5938:
5935:
5932:
5929:
5927:
5924:
5922:
5919:
5918:
5916:
5905:
5899:
5896:
5894:
5891:
5889:
5886:
5885:
5882:
5879:
5877:
5874:
5872:
5869:
5868:
5865:
5862:
5860:
5857:
5855:
5852:
5851:
5849:
5840:
5837:
5830:
5821:
5815:
5812:
5810:
5807:
5805:
5802:
5801:
5798:
5795:
5793:
5790:
5788:
5785:
5784:
5781:
5778:
5775:
5773:
5770:
5768:
5765:
5764:
5762:
5751:
5745:
5742:
5740:
5737:
5735:
5732:
5731:
5728:
5725:
5723:
5720:
5718:
5715:
5714:
5711:
5708:
5705:
5703:
5700:
5698:
5695:
5694:
5692:
5683:
5680:
5673:
5664:
5658:
5655:
5653:
5650:
5648:
5645:
5644:
5641:
5638:
5636:
5633:
5631:
5628:
5627:
5624:
5621:
5618:
5616:
5613:
5611:
5608:
5607:
5605:
5594:
5588:
5585:
5583:
5580:
5578:
5575:
5574:
5571:
5568:
5566:
5563:
5561:
5558:
5557:
5554:
5551:
5548:
5546:
5543:
5541:
5538:
5537:
5535:
5526:
5523:
5509:
5508:
5493:
5489:
5487:
5483:
5481:
5479:
5476:
5472:
5470:
5466:
5464:
5462:
5459:
5455:
5453:
5449:
5447:
5445:
5442:
5439:
5438:
5435:
5431:
5429:
5425:
5423:
5421:
5418:
5414:
5412:
5408:
5406:
5404:
5401:
5397:
5395:
5391:
5389:
5387:
5384:
5381:
5380:
5377:
5373:
5371:
5367:
5365:
5363:
5360:
5356:
5354:
5350:
5348:
5346:
5343:
5339:
5337:
5333:
5331:
5329:
5326:
5325:
5300:Main article:
5297:
5294:
5269:
5268:
5253:
5249:
5242:
5239:
5237:
5234:
5232:
5229:
5227:
5224:
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5220:
5217:
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5212:
5210:
5207:
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5201:
5198:
5195:
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5190:
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5180:
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5176:
5173:
5169:
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5158:
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5145:
5143:
5140:
5139:
5136:
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5117:
5114:
5111:
5109:
5106:
5104:
5101:
5099:
5096:
5095:
5092:
5088:
5084:
5077:
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5069:
5067:
5064:
5062:
5059:
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5017:
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5012:
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5008:
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5000:
4993:
4990:
4988:
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4960:
4958:
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4914:
4912:
4910:
4906:
4899:
4896:
4893:
4891:
4888:
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4880:
4878:
4875:
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4710:
4707:
4705:
4702:
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4697:
4695:
4692:
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4688:
4685:
4682:
4680:
4677:
4675:
4672:
4669:
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4482:
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4464:
4457:
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4449:
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4439:
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4435:
4432:
4430:
4427:
4425:
4422:
4420:
4417:
4416:
4413:
4410:
4408:
4405:
4402:
4400:
4397:
4395:
4392:
4391:
4388:
4358:Main article:
4355:
4352:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4263:
4262:
4247:
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4241:
4238:
4235:
4234:
4231:
4228:
4225:
4219:
4216:
4210:
4207:
4204:
4201:
4198:
4195:
4194:
4176:
4175:
4162:
4157:
4154:
4151:
4145:
4142:
4136:
4133:
4130:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4107:
4105:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4041:
4040:
4027:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
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3924:
3908:
3907:
3904:
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3898:
3890:
3887:
3871:
3870:
3852:
3849:
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3841:
3836:
3833:
3828:
3825:
3822:
3804:
3801:
3796:
3793:
3788:
3785:
3780:
3777:
3774:
3714:
3713:
3697:
3694:
3691:
3688:
3685:
3682:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3634:
3633:
3617:
3614:
3610:
3606:
3603:
3600:
3597:
3593:
3589:
3586:
3583:
3580:
3576:
3572:
3569:
3566:
3563:
3562:
3558:
3555:
3551:
3547:
3544:
3541:
3538:
3534:
3530:
3527:
3524:
3521:
3517:
3513:
3510:
3507:
3506:
3461:
3458:
3454:
3451:
3448:
3444:
3441:
3438:
3410:
3407:
3404:
3401:
3397:
3394:
3391:
3388:
3385:
3381:
3378:
3375:
3372:
3369:
3356:
3353:
3340:
3337:
3327:
3324:
3268:
3267:
3251:
3248:
3244:
3240:
3237:
3234:
3231:
3227:
3223:
3220:
3217:
3214:
3213:
3209:
3206:
3202:
3198:
3195:
3192:
3188:
3184:
3181:
3178:
3175:
3174:
3170:
3167:
3163:
3159:
3156:
3153:
3149:
3145:
3142:
3139:
3138:
3109:
3108:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
2986:The equations
2973:
2970:
2965:
2964:
2948:
2945:
2941:
2937:
2934:
2931:
2928:
2924:
2920:
2917:
2914:
2911:
2910:
2906:
2903:
2899:
2895:
2892:
2889:
2886:
2882:
2878:
2875:
2872:
2869:
2868:
2864:
2861:
2858:
2854:
2850:
2847:
2844:
2841:
2837:
2833:
2830:
2827:
2826:
2809:
2808:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2694:The equations
2675:
2672:
2670:
2667:
2659:, or if it is
2643:
2642:
2639:
2638:
2635:
2634:Two equations
2632:
2628:
2627:
2620:
2613:
2573:
2572:
2565:
2562:
2541:
2538:
2468:
2465:
2464:
2463:
2456:
2449:
2414:
2410:
2406:
2403:
2400:
2395:
2391:
2387:
2382:
2378:
2328:
2325:
2306:
2301:
2293:
2289:
2285:
2284:
2281:
2278:
2277:
2272:
2268:
2264:
2263:
2258:
2254:
2250:
2249:
2247:
2242:
2238:
2233:
2228:
2220:
2216:
2212:
2211:
2208:
2205:
2204:
2199:
2195:
2191:
2190:
2185:
2181:
2177:
2176:
2174:
2169:
2165:
2160:
2155:
2147:
2144:
2140:
2136:
2134:
2131:
2127:
2124:
2120:
2116:
2112:
2109:
2105:
2101:
2100:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2078:
2073:
2070:
2066:
2062:
2060:
2057:
2053:
2049:
2045:
2041:
2037:
2033:
2032:
2027:
2024:
2020:
2016:
2014:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1986:
1984:
1979:
1976:
1921:
1917:
1913:
1909:
1893:
1890:
1840:
1839:
1826:
1818:
1814:
1810:
1809:
1806:
1803:
1802:
1797:
1793:
1789:
1788:
1783:
1779:
1775:
1774:
1772:
1767:
1762:
1754:
1751:
1747:
1743:
1742:
1739:
1736:
1735:
1730:
1727:
1723:
1719:
1718:
1713:
1710:
1706:
1702:
1701:
1699:
1692:
1688:
1684:
1681:
1678:
1673:
1665:
1662:
1658:
1654:
1653:
1650:
1647:
1646:
1641:
1637:
1633:
1632:
1627:
1623:
1619:
1618:
1616:
1609:
1605:
1601:
1596:
1588:
1585:
1581:
1577:
1576:
1573:
1570:
1569:
1564:
1560:
1556:
1555:
1550:
1546:
1542:
1541:
1539:
1532:
1528:
1504:
1501:
1463:
1459:
1455:
1452:
1449:
1444:
1440:
1436:
1431:
1427:
1404:
1401:
1397:
1393:
1390:
1387:
1382:
1378:
1374:
1369:
1365:
1342:
1338:
1334:
1331:
1328:
1323:
1319:
1315:
1310:
1306:
1294:
1293:
1280:
1275:
1270:
1266:
1262:
1257:
1253:
1247:
1244:
1240:
1236:
1233:
1230:
1225:
1221:
1215:
1212:
1208:
1204:
1199:
1195:
1189:
1186:
1182:
1178:
1177:
1174:
1171:
1170:
1165:
1161:
1157:
1152:
1148:
1142:
1139:
1135:
1131:
1128:
1125:
1120:
1116:
1110:
1106:
1102:
1097:
1093:
1087:
1083:
1079:
1078:
1073:
1069:
1065:
1060:
1056:
1050:
1047:
1043:
1039:
1036:
1033:
1028:
1024:
1018:
1014:
1010:
1005:
1001:
995:
991:
987:
986:
984:
952:
949:
930:
927:
922:
919:
899:
879:
876:
873:
853:
842:
841:
830:
827:
824:
821:
818:
814:
810:
805:
802:
797:
794:
790:
786:
764:
763:
752:
749:
744:
741:
736:
733:
730:
727:
704:
684:
673:
672:
657:
654:
652:
650:
647:
643:
639:
636:
633:
630:
626:
622:
619:
616:
613:
612:
608:
605:
601:
597:
594:
591:
588:
584:
580:
577:
574:
571:
570:
555:
552:
548:
547:
536:
533:
530:
516:
515:
504:
501:
498:
495:
480:
477:
475:
472:
446:, such as the
417:complex system
365:linear algebra
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
288:ordered triple
264:
263:
250:
245:
242:
239:
236:
233:
228:
225:
220:
217:
214:
211:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
174:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
143:
142:
140:
90:
89:
44:
42:
35:
26:
9:
6:
4:
3:
2:
9085:
9074:
9071:
9069:
9066:
9064:
9061:
9060:
9058:
9043:
9035:
9034:
9031:
9025:
9022:
9020:
9019:Sparse matrix
9017:
9015:
9012:
9010:
9007:
9005:
9002:
9001:
8999:
8997:
8993:
8987:
8984:
8982:
8979:
8977:
8974:
8972:
8969:
8967:
8964:
8962:
8959:
8958:
8956:
8954:constructions
8953:
8949:
8943:
8942:Outermorphism
8940:
8938:
8935:
8933:
8930:
8928:
8925:
8923:
8920:
8918:
8915:
8913:
8910:
8908:
8905:
8903:
8902:Cross product
8900:
8898:
8895:
8894:
8892:
8890:
8886:
8880:
8877:
8875:
8872:
8870:
8869:Outer product
8867:
8865:
8862:
8860:
8857:
8855:
8852:
8850:
8849:Orthogonality
8847:
8846:
8844:
8842:
8838:
8832:
8829:
8827:
8826:Cramer's rule
8824:
8822:
8819:
8817:
8814:
8812:
8809:
8807:
8804:
8802:
8799:
8797:
8796:Decomposition
8794:
8792:
8789:
8788:
8786:
8784:
8780:
8775:
8765:
8762:
8760:
8757:
8755:
8752:
8750:
8747:
8745:
8742:
8740:
8737:
8735:
8732:
8730:
8727:
8725:
8722:
8720:
8717:
8715:
8712:
8710:
8707:
8705:
8702:
8700:
8697:
8695:
8692:
8690:
8687:
8685:
8682:
8680:
8677:
8675:
8672:
8671:
8669:
8665:
8659:
8656:
8654:
8651:
8650:
8647:
8643:
8636:
8631:
8629:
8624:
8622:
8617:
8616:
8613:
8606:
8601:
8597:
8596:
8586:
8582:
8577:
8572:
8568:
8564:
8559:
8555:
8551:
8547:
8543:
8538:
8534:
8529:
8525:
8523:0-534-99845-3
8519:
8515:
8510:
8506:
8502:
8496:
8492:
8491:
8485:
8481:
8475:
8471:
8466:
8462:
8460:0-387-98259-0
8456:
8452:
8447:
8446:
8437:
8435:0-7514-0159-5
8431:
8427:
8422:
8419:
8413:
8409:
8404:
8401:
8397:
8393:
8389:
8385:
8381:
8377:
8373:
8368:
8363:
8359:
8355:
8350:
8347:
8345:0-13-487538-9
8341:
8337:
8336:Prentice-Hall
8333:
8328:
8325:
8323:0-8018-5414-8
8319:
8315:
8311:
8306:
8303:
8297:
8294:, MA: Dover,
8293:
8288:
8285:
8283:0-534-93219-3
8279:
8275:
8270:
8269:
8262:
8259:
8257:0-395-14017-X
8253:
8249:
8244:
8243:
8236:
8233:
8231:0-471-84819-0
8227:
8223:
8219:
8214:
8213:
8202:
8197:
8189:
8183:
8168:
8167:
8162:
8155:
8148:
8144:
8139:
8133:, p. 68.
8132:
8127:
8125:
8117:
8113:
8108:
8101:
8100:Cullen (1990)
8096:
8088:
8081:
8075:
8068:
8063:
8057:, p. 65.
8056:
8051:
8036:
8032:
8026:
8020:, p. 57.
8019:
8018:Harper (1976)
8015:
8011:
8008:, p. 2;
8007:
8002:
7998:
7988:
7985:
7983:
7980:
7977:
7974:
7972:
7969:
7967:
7964:
7962:
7959:
7957:
7954:
7951:
7947:
7943:
7939:
7936:
7934:
7931:
7929:
7926:
7924:
7921:
7920:
7913:
7911:
7908:
7904:
7900:
7895:
7891:
7888:
7882:
7880:
7876:
7872:
7867:
7863:
7860:
7855:
7850:
7846:
7843:
7823:
7819:
7810:
7802:
7789:
7781:
7772:
7764:
7763:
7762:
7759:
7755:
7752:
7747:
7728:
7720:
7712:
7697:
7689:
7682:
7681:
7680:
7672:
7670:
7666:
7662:
7658:
7650:
7647:
7643:
7639:
7635:
7631:
7627:
7623:
7618:
7614:
7610:
7606:
7605:
7604:
7600:
7594:
7590:
7586:
7576:
7574:
7570:
7566:
7563:entries, and
7562:
7558:
7553:
7549:
7544:
7520:
7512:
7505:
7504:
7503:
7482:
7474:
7465:
7461:
7455:
7452:
7448:
7440:
7437:
7434:
7425:
7421:
7415:
7412:
7408:
7400:
7391:
7387:
7381:
7378:
7374:
7359:
7342:
7334:
7325:
7321:
7315:
7312:
7308:
7300:
7297:
7294:
7285:
7281:
7275:
7271:
7263:
7254:
7250:
7244:
7240:
7232:
7224:
7215:
7211:
7205:
7202:
7198:
7190:
7187:
7184:
7175:
7171:
7165:
7161:
7153:
7144:
7140:
7134:
7130:
7118:
7117:
7116:
7114:
7108:
7098:
7096:
7091:
7089:
7068:
7065:
7062:
7025:
6982:
6964:
6961:
6958:
6952:
6937:
6934:
6930:
6926:
6918:
6915:
6912:
6893:
6892:
6891:
6872:
6840:
6837:
6834:
6814:
6794:
6786:
6785:Jacobi method
6782:
6776:
6774:
6770:
6766:
6762:
6758:
6755:
6751:
6746:
6744:
6740:
6736:
6732:
6728:
6727:
6722:
6716:
6709:Other methods
6706:
6616:
6613:
6609:
6605:
6596:
6592:
6589:
6586:
6582:
6578:
6568:
6565:
6561:
6557:
6548:
6544:
6539:
6536:
6532:
6528:
6525:
6521:
6517:
6507:
6504:
6500:
6496:
6484:
6483:
6482:
6466:
6463:
6459:
6436:
6432:
6423:
6406:
6398:
6388:
6384:
6380:
6355:
6347:
6322:
6309:
6262:
6258:
6253:
6249:
6245:
6242:
6238:
6234:
6224:
6220:
6216:
6204:
6203:
6202:
6186:
6182:
6173:
6169:
6165:
6161:
6157:
6153:
6149:
6131:
6128:
6124:
6093:
6090:
6086:
6082:
6070:
6069:
6068:
6066:
6062:
6058:
6054:
6050:
6029:
6021:
6007:
6003:
6001:
5982:
5973:
5967:
5962:
5957:
5950:
5945:
5940:
5933:
5930:
5925:
5920:
5914:
5903:
5897:
5892:
5887:
5880:
5875:
5870:
5863:
5858:
5853:
5847:
5838:
5835:
5828:
5819:
5813:
5808:
5803:
5796:
5791:
5786:
5779:
5776:
5771:
5766:
5760:
5749:
5743:
5738:
5733:
5726:
5721:
5716:
5709:
5706:
5701:
5696:
5690:
5681:
5678:
5671:
5662:
5656:
5651:
5646:
5639:
5634:
5629:
5622:
5619:
5614:
5609:
5603:
5592:
5586:
5581:
5576:
5569:
5564:
5559:
5552:
5549:
5544:
5539:
5533:
5524:
5521:
5514:
5513:
5512:
5491:
5485:
5482:
5477:
5474:
5468:
5465:
5460:
5457:
5451:
5448:
5443:
5440:
5433:
5427:
5424:
5419:
5416:
5410:
5407:
5402:
5399:
5393:
5390:
5385:
5382:
5375:
5369:
5366:
5361:
5358:
5352:
5349:
5344:
5341:
5335:
5332:
5327:
5316:
5315:
5314:
5312:
5308:
5307:Cramer's rule
5303:
5302:Cramer's rule
5296:Cramer's rule
5293:
5289:
5282:
5275:
5251:
5247:
5240:
5235:
5230:
5225:
5218:
5213:
5208:
5203:
5196:
5193:
5188:
5183:
5178:
5171:
5167:
5163:
5156:
5151:
5146:
5141:
5134:
5129:
5124:
5119:
5112:
5107:
5102:
5097:
5090:
5086:
5082:
5075:
5070:
5065:
5060:
5053:
5048:
5043:
5038:
5031:
5026:
5023:
5018:
5013:
5006:
5002:
4998:
4991:
4986:
4981:
4976:
4969:
4964:
4961:
4956:
4951:
4944:
4939:
4936:
4931:
4926:
4919:
4915:
4913:
4904:
4897:
4894:
4889:
4884:
4881:
4876:
4869:
4864:
4861:
4856:
4851:
4844:
4839:
4836:
4831:
4826:
4819:
4815:
4811:
4804:
4801:
4796:
4791:
4788:
4783:
4776:
4773:
4768:
4763:
4760:
4755:
4748:
4743:
4740:
4735:
4730:
4723:
4719:
4715:
4708:
4703:
4698:
4693:
4686:
4683:
4678:
4673:
4670:
4665:
4658:
4653:
4650:
4645:
4640:
4633:
4629:
4627:
4621:
4614:
4609:
4604:
4599:
4592:
4587:
4582:
4577:
4570:
4565:
4562:
4557:
4552:
4545:
4533:
4532:
4531:
4529:
4525:
4520:
4514:
4511:
4508:
4504:
4501:
4498:
4495:
4494:
4493:
4491:
4487:
4462:
4455:
4450:
4445:
4440:
4433:
4428:
4423:
4418:
4411:
4406:
4403:
4398:
4393:
4386:
4378:
4377:
4376:
4375:
4371:
4367:
4366:row reduction
4361:
4354:Row reduction
4351:
4334:
4331:
4328:
4325:
4322:
4319:
4313:
4307:
4304:
4301:
4298:
4295:
4284:
4280:
4276:
4272:
4268:
4265:Substituting
4245:
4242:
4239:
4229:
4226:
4223:
4217:
4214:
4208:
4205:
4202:
4199:
4196:
4185:
4184:
4183:
4181:
4155:
4152:
4149:
4143:
4140:
4134:
4131:
4124:
4121:
4118:
4115:
4112:
4109:
4103:
4094:
4093:
4092:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4046:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3922:
3913:
3912:
3911:
3905:
3902:
3899:
3896:
3895:
3894:
3886:
3884:
3880:
3876:
3850:
3847:
3842:
3839:
3834:
3831:
3826:
3823:
3820:
3802:
3799:
3794:
3791:
3786:
3783:
3778:
3775:
3772:
3765:
3764:
3763:
3760:
3758:
3754:
3750:
3745:
3743:
3739:
3735:
3731:
3727:
3723:
3719:
3695:
3692:
3689:
3686:
3683:
3680:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3639:
3638:
3637:
3615:
3608:
3601:
3598:
3591:
3584:
3581:
3574:
3567:
3564:
3556:
3549:
3542:
3539:
3532:
3525:
3522:
3515:
3508:
3497:
3496:
3495:
3492:
3490:
3486:
3482:
3478:
3473:
3456:
3452:
3449:
3446:
3442:
3439:
3428:
3424:
3405:
3402:
3399:
3395:
3392:
3389:
3386:
3383:
3379:
3376:
3373:
3370:
3352:
3350:
3346:
3336:
3333:
3323:
3321:
3316:
3312:
3308:
3304:
3300:
3296:
3291:
3287:
3279:
3275:
3249:
3242:
3235:
3232:
3225:
3218:
3215:
3207:
3200:
3193:
3186:
3179:
3176:
3168:
3161:
3154:
3147:
3140:
3129:
3128:
3127:
3124:
3122:
3118:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3036:
3035:
3034:
3031:
3025:
3024:contradiction
3021:
3017:
3007:
3003:
2995:
2991:
2984:
2979:
2969:
2946:
2939:
2932:
2929:
2922:
2915:
2912:
2904:
2897:
2890:
2887:
2880:
2873:
2870:
2862:
2859:
2852:
2845:
2842:
2835:
2828:
2817:
2816:
2815:
2812:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2736:
2735:
2734:
2726:
2722:
2714:
2710:
2702:
2698:
2692:
2688:
2686:
2681:
2666:
2664:
2663:
2658:
2657:
2651:
2647:
2631:One equation
2629:
2625:
2618:
2611:
2606:
2603:
2602:
2601:
2598:
2596:
2592:
2587:
2583:
2578:
2570:
2566:
2563:
2560:
2556:
2555:
2554:
2546:
2537:
2535:
2531:
2527:
2525:
2520:
2516:
2511:
2509:
2505:
2500:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2461:
2457:
2454:
2450:
2447:
2443:
2442:
2441:
2438:
2436:
2435:
2430:
2412:
2408:
2404:
2401:
2398:
2393:
2389:
2385:
2380:
2376:
2367:
2366:
2355:
2351:
2343:
2339:
2333:
2324:
2322:
2321:
2304:
2299:
2291:
2287:
2279:
2270:
2266:
2256:
2252:
2245:
2240:
2231:
2226:
2218:
2214:
2206:
2197:
2193:
2183:
2179:
2172:
2167:
2158:
2153:
2145:
2142:
2138:
2132:
2125:
2122:
2118:
2110:
2107:
2103:
2095:
2090:
2085:
2080:
2071:
2068:
2064:
2058:
2051:
2047:
2039:
2035:
2025:
2022:
2018:
2012:
2005:
2001:
1993:
1989:
1982:
1977:
1974:
1966:
1964:
1960:
1957:entries, and
1956:
1952:
1951:column vector
1948:
1944:
1940:
1936:
1915:
1907:
1899:
1889:
1887:
1883:
1879:
1875:
1874:
1869:
1865:
1864:
1859:
1858:
1853:
1852:
1847:
1846:
1845:vector spaces
1824:
1816:
1812:
1804:
1795:
1791:
1781:
1777:
1770:
1765:
1760:
1752:
1749:
1745:
1737:
1728:
1725:
1721:
1711:
1708:
1704:
1697:
1690:
1686:
1682:
1679:
1676:
1671:
1663:
1660:
1656:
1648:
1639:
1635:
1625:
1621:
1614:
1607:
1603:
1599:
1594:
1586:
1583:
1579:
1571:
1562:
1558:
1548:
1544:
1537:
1530:
1526:
1518:
1517:
1516:
1514:
1510:
1509:column vector
1500:
1498:
1494:
1490:
1486:
1482:
1477:
1461:
1457:
1453:
1450:
1447:
1442:
1438:
1434:
1429:
1425:
1402:
1399:
1395:
1391:
1388:
1385:
1380:
1376:
1372:
1367:
1363:
1340:
1336:
1332:
1329:
1326:
1321:
1317:
1313:
1308:
1304:
1273:
1268:
1264:
1260:
1255:
1251:
1245:
1242:
1238:
1234:
1231:
1228:
1223:
1219:
1213:
1210:
1206:
1202:
1197:
1193:
1187:
1184:
1180:
1172:
1163:
1159:
1155:
1150:
1146:
1140:
1137:
1133:
1129:
1126:
1123:
1118:
1114:
1108:
1104:
1100:
1095:
1091:
1085:
1081:
1071:
1067:
1063:
1058:
1054:
1048:
1045:
1041:
1037:
1034:
1031:
1026:
1022:
1016:
1012:
1008:
1003:
999:
993:
989:
982:
973:
972:
971:
969:
965:
962:
958:
948:
946:
928:
925:
920:
917:
897:
877:
874:
871:
851:
828:
825:
822:
819:
816:
812:
808:
803:
800:
795:
792:
788:
784:
777:
776:
775:
773:
769:
750:
747:
742:
739:
734:
731:
728:
725:
718:
717:
716:
702:
682:
655:
653:
648:
641:
634:
631:
624:
617:
614:
606:
599:
592:
589:
582:
575:
572:
561:
560:
559:
551:
534:
531:
528:
521:
520:
519:
502:
499:
496:
493:
486:
485:
484:
471:
469:
465:
461:
460:Gröbner basis
457:
453:
449:
445:
441:
437:
433:
429:
425:
420:
418:
414:
410:
406:
405:linearization
402:
399:can often be
398:
394:
390:
386:
382:
378:
374:
370:
366:
361:
347:
341:
338:
335:
332:
329:
326:
323:
317:
311:
308:
305:
302:
299:
289:
285:
284:
278:
274:
270:
243:
240:
237:
234:
231:
226:
223:
218:
215:
212:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
168:
165:
162:
159:
156:
153:
150:
147:
144:
138:
129:
128:
127:
125:
121:
117:
116:linear system
113:
109:
101:
96:
86:
83:
75:
65:
61:
55:
54:
48:
43:
34:
33:
30:
19:
8952:Vector space
8763:
8684:Vector space
8569:(7): 79–86.
8566:
8562:
8553:
8541:
8532:
8513:
8505:the original
8489:
8469:
8450:
8425:
8407:
8357:
8353:
8331:
8309:
8291:
8267:
8241:
8217:
8210:Bibliography
8196:
8182:
8170:. Retrieved
8164:
8154:
8138:
8107:
8102:, p. 3.
8095:
8086:
8074:
8062:
8050:
8038:. Retrieved
8034:
8025:
8006:Anton (1987)
8001:
7933:Coates graph
7909:
7901:lies in the
7898:
7893:
7889:
7886:
7883:
7878:
7865:
7861:
7858:
7848:
7844:
7841:
7838:
7757:
7753:
7750:
7745:
7743:
7678:
7668:
7660:
7654:
7648:
7645:
7637:
7633:
7625:
7621:
7612:
7608:
7598:
7588:
7584:
7582:
7572:
7564:
7560:
7556:
7551:
7547:
7542:
7540:
7501:
7112:
7110:
7092:
7087:
7005:
6777:
6749:
6747:
6742:
6738:
6734:
6725:
6718:
6638:
6421:
6307:
6283:
6171:
6163:
6159:
6155:
6151:
6115:
6064:
6060:
6056:
6052:
6048:
6013:
6004:
5997:
5511:is given by
5510:
5311:determinants
5306:
5305:
5287:
5280:
5273:
5270:
4521:
4518:
4512:
4502:
4496:
4483:
4369:
4365:
4363:
4282:
4278:
4274:
4270:
4266:
4264:
4179:
4177:
4044:
4042:
3909:
3892:
3882:
3878:
3874:
3872:
3761:
3756:
3746:
3741:
3737:
3733:
3729:
3725:
3721:
3717:
3715:
3635:
3493:
3488:
3484:
3480:
3476:
3474:
3358:
3342:
3331:
3329:
3314:
3310:
3292:
3288:
3277:
3273:
3269:
3125:
3116:
3110:
3032:
3019:
3016:inconsistent
3015:
3013:
3005:
3001:
2993:
2989:
2966:
2813:
2810:
2732:
2724:
2720:
2712:
2708:
2700:
2696:
2679:
2677:
2674:Independence
2662:inconsistent
2660:
2654:
2652:
2648:
2644:
2599:
2594:
2590:
2585:
2581:
2574:
2551:
2533:
2523:
2514:
2512:
2501:
2493:intersection
2484:
2476:
2472:
2470:
2459:
2452:
2445:
2439:
2434:solution set
2432:
2363:
2361:
2353:
2349:
2341:
2337:
2327:Solution set
2318:
1967:
1962:
1958:
1954:
1946:
1942:
1938:
1934:
1895:
1885:
1881:
1877:
1871:
1861:
1855:
1849:
1843:
1841:
1506:
1478:
1295:
968:coefficients
960:
956:
954:
951:General form
843:
771:
765:
695:in terms of
674:
557:
549:
517:
482:
438:. For other
424:coefficients
421:
401:approximated
362:
281:
276:
272:
268:
265:
115:
111:
105:
78:
72:October 2015
69:
50:
29:
8932:Multivector
8897:Determinant
8854:Dot product
8699:Linear span
7854:translation
7569:zero vector
7113:homogeneous
5276:= −15
3481:independent
3326:Equivalence
2972:Consistency
2680:independent
2460:no solution
377:engineering
108:mathematics
64:introducing
9057:Categories
8966:Direct sum
8801:Invertible
8704:Linear map
8576:2007.10254
8246:, Boston:
8145:, p.
8114:, p.
8040:August 26,
8035:Britannica
7994:References
7665:null space
7105:See also:
6174:, denoted
3485:parameters
3345:algorithms
3332:equivalent
3020:consistent
2976:See also:
2703:= −1
2669:Properties
2519:hyperplane
2344:= −1
768:substitute
369:algorithms
47:references
9063:Equations
8996:Numerical
8759:Transpose
8563:Comm. ACM
8367:0811.3171
7575:entries.
7438:⋯
7360:⋮
7298:⋯
7188:⋯
7088:converged
6953:−
6935:−
6754:symmetric
6721:Cracovian
6614:−
6590:−
6566:−
6537:−
6529:−
6505:−
6464:−
6246:−
6129:−
6091:−
6055:rows and
5931:−
5777:−
5707:−
5620:−
5550:−
5353:−
5194:−
5168:∼
5087:∼
5024:−
5003:∼
4962:−
4937:−
4916:∼
4895:−
4882:−
4862:−
4837:−
4816:∼
4802:−
4789:−
4774:−
4761:−
4741:−
4720:∼
4684:−
4671:−
4651:−
4630:∼
4563:−
4404:−
4320:−
4237:⇒
4073:−
3941:−
3843:−
3827:−
3779:−
3753:dimension
3662:−
3653:−
3533:−
3489:dependent
3447:−
3390:−
2860:−
2836:−
2699:− 2
2577:dimension
2497:empty set
2402:…
2280:⋮
2207:⋮
2133:⋯
2096:⋮
2091:⋱
2086:⋮
2081:⋮
2059:⋯
2013:⋯
1965:entries.
1873:dimension
1805:⋮
1738:⋮
1680:⋯
1649:⋮
1572:⋮
1451:…
1389:…
1330:…
1232:⋯
1173:⋮
1127:⋯
1035:⋯
796:−
735:−
393:economics
385:chemistry
339:−
330:−
235:−
213:−
203:−
182:−
160:−
124:variables
9042:Category
8981:Subspace
8976:Quotient
8927:Bivector
8841:Bilinear
8783:Matrices
8658:Glossary
8552:(2005).
8392:19905613
8172:March 9,
7916:See also
7615:are two
7555:matrix,
6726:pivoting
6340:satisfy
3483:, or as
3121:parallel
2589:, where
2584:−
2365:solution
2340:−
1945:matrix,
1489:integers
964:unknowns
452:integers
283:solution
8653:Outline
8400:5187993
8372:Bibcode
7942:Fortran
7905:of the
7644:, then
7640:is any
7617:vectors
7589:trivial
7567:is the
6148:inverse
6146:is the
3349:solving
3301:of the
3123:lines.
2968:point.
2483:on the
1851:modules
910:yields
381:physics
60:improve
8937:Tensor
8749:Kernel
8679:Vector
8674:Scalar
8520:
8497:
8476:
8457:
8432:
8414:
8398:
8390:
8342:
8320:
8298:
8280:
8254:
8228:
7938:LAPACK
7642:scalar
7545:is an
7541:where
7364:
6284:where
6116:where
4513:Type 3
4507:scalar
4503:Type 2
4497:Type 1
4047:gives
3427:vector
2717:, and
1937:is an
1933:where
1898:matrix
1487:, but
1296:where
454:, see
391:, and
100:planes
49:, but
8806:Minor
8791:Block
8729:Basis
8571:arXiv
8396:S2CID
8362:arXiv
8222:Wiley
8083:(PDF)
7903:image
7852:is a
7601:) ≠ 0
7571:with
3873:Here
3716:Here
3284:0 = 1
3113:0 = 1
3028:0 = 1
2504:plane
2489:plane
1953:with
1949:is a
1863:basis
1511:in a
436:field
8961:Dual
8816:Rank
8518:ISBN
8495:ISBN
8474:ISBN
8455:ISBN
8430:ISBN
8412:ISBN
8388:PMID
8340:ISBN
8318:ISBN
8296:ISBN
8278:ISBN
8252:ISBN
8226:ISBN
8174:2021
8042:2024
7871:flat
7611:and
7597:det(
7587:(or
7585:zero
7043:and
4526:and
4277:and
3881:and
3740:and
3724:and
3479:(or
3477:free
3347:for
3299:rank
3008:= 12
2998:and
2530:flat
2513:For
2481:line
2475:and
2346:and
2320:rank
1857:span
1491:and
1481:real
966:and
766:Now
448:ring
428:real
395:. A
280:. A
114:(or
110:, a
8581:doi
8380:doi
8358:103
8147:235
7950:C++
7708:and
7659:of
7632:If
7607:If
6170:of
6150:of
5290:= 2
5283:= 8
4364:In
3813:and
3673:and
3280:= 2
3276:+ 2
3070:and
3004:+ 2
2996:= 6
2992:+ 2
2770:and
2727:= 7
2723:+ 3
2715:= 8
2711:+ 5
2521:in
2506:in
2429:set
2356:= 9
1880:or
1866:of
1483:or
947:.)
829:15.
450:of
430:or
411:or
106:In
9059::
8579:.
8567:67
8565:.
8394:,
8386:,
8378:,
8370:,
8356:,
8338:,
8316:,
8276:,
8250:,
8224:,
8163:.
8123:^
8116:70
8085:.
8033:.
7948:,
7944:,
7912:.
7892:=
7881:.
7864:=
7847:=
7756:=
7671:.
7624:+
7550:×
7483:0.
7276:22
7245:21
7166:12
7135:11
7097:.
6763:.
6745:.
5285:,
5278:,
5197:15
4769:12
4679:12
4350:.
4323:15
3800:11
3744:.
3117:xy
3096:12
3030:.
2796:12
2705:,
2687:.
2536:.
2499:.
2485:xy
2437:.
2362:A
2352:+
2052:22
2040:21
2006:12
1994:11
1640:22
1626:12
1563:21
1549:11
1515:.
1499:.
1381:12
1368:11
1109:22
1086:21
1017:12
994:11
715::
649:15
535:2.
470:.
419:.
387:,
383:,
379:,
275:,
271:,
8634:e
8627:t
8620:v
8587:.
8583::
8573::
8556:.
8526:.
8482:.
8463:.
8382::
8374::
8364::
8203:.
8190:.
8176:.
8149:.
8118:.
8089:.
8044:.
7952:)
7946:C
7910:A
7899:b
7894:b
7890:x
7887:A
7879:p
7866:0
7862:x
7859:A
7849:b
7845:x
7842:A
7824:.
7820:}
7815:0
7811:=
7807:x
7803:A
7794:v
7790::
7786:v
7782:+
7778:p
7773:{
7758:b
7754:x
7751:A
7746:p
7729:.
7725:0
7721:=
7717:x
7713:A
7702:b
7698:=
7694:x
7690:A
7669:A
7661:R
7649:u
7646:r
7638:r
7634:u
7626:v
7622:u
7613:v
7609:u
7599:A
7595:(
7573:m
7565:0
7561:n
7557:x
7552:n
7548:m
7543:A
7525:0
7521:=
7517:x
7513:A
7475:=
7466:n
7462:x
7456:n
7453:m
7449:a
7441:+
7435:+
7426:2
7422:x
7416:2
7413:m
7409:a
7401:+
7392:1
7388:x
7382:1
7379:m
7375:a
7343:0
7335:=
7326:n
7322:x
7316:n
7313:2
7309:a
7301:+
7295:+
7286:2
7282:x
7272:a
7264:+
7255:1
7251:x
7241:a
7233:0
7225:=
7216:n
7212:x
7206:n
7203:1
7199:a
7191:+
7185:+
7176:2
7172:x
7162:a
7154:+
7145:1
7141:x
7131:a
7072:)
7069:1
7066:+
7063:k
7060:(
7054:x
7029:)
7026:k
7023:(
7017:x
6991:)
6986:)
6983:k
6980:(
6974:x
6968:)
6965:U
6962:+
6959:L
6956:(
6948:b
6943:(
6938:1
6931:D
6927:=
6922:)
6919:1
6916:+
6913:k
6910:(
6904:x
6876:)
6873:0
6870:(
6864:x
6841:U
6838:+
6835:L
6815:D
6795:A
6750:A
6743:b
6739:A
6735:A
6692:w
6670:w
6648:w
6623:b
6617:1
6610:A
6606:=
6602:w
6597:)
6593:I
6587:I
6583:(
6579:+
6575:b
6569:1
6562:A
6558:=
6554:w
6549:)
6545:A
6540:1
6533:A
6526:I
6522:(
6518:+
6514:b
6508:1
6501:A
6497:=
6493:x
6467:1
6460:A
6437:+
6433:A
6422:A
6407:.
6403:b
6399:=
6395:b
6389:+
6385:A
6381:A
6360:b
6356:=
6352:x
6348:A
6327:0
6323:=
6319:w
6308:n
6293:w
6268:w
6263:)
6259:A
6254:+
6250:A
6243:I
6239:(
6235:+
6231:b
6225:+
6221:A
6217:=
6213:x
6187:+
6183:A
6172:A
6164:A
6160:n
6158:=
6156:m
6152:A
6132:1
6125:A
6100:b
6094:1
6087:A
6083:=
6079:x
6065:m
6061:m
6059:=
6057:n
6053:m
6049:A
6034:b
6030:=
6026:x
6022:A
5983:.
5974:|
5968:3
5963:4
5958:2
5951:6
5946:5
5941:3
5934:2
5926:3
5921:1
5915:|
5904:|
5898:8
5893:4
5888:2
5881:7
5876:5
5871:3
5864:5
5859:3
5854:1
5848:|
5839:=
5836:z
5829:,
5820:|
5814:3
5809:4
5804:2
5797:6
5792:5
5787:3
5780:2
5772:3
5767:1
5761:|
5750:|
5744:3
5739:8
5734:2
5727:6
5722:7
5717:3
5710:2
5702:5
5697:1
5691:|
5682:=
5679:y
5672:,
5663:|
5657:3
5652:4
5647:2
5640:6
5635:5
5630:3
5623:2
5615:3
5610:1
5604:|
5593:|
5587:3
5582:4
5577:8
5570:6
5565:5
5560:7
5553:2
5545:3
5540:5
5534:|
5525:=
5522:x
5492:8
5486:=
5478:z
5475:3
5469:+
5461:y
5458:4
5452:+
5444:x
5441:2
5434:7
5428:=
5420:z
5417:6
5411:+
5403:y
5400:5
5394:+
5386:x
5383:3
5376:5
5370:=
5362:z
5359:2
5345:y
5342:3
5336:+
5328:x
5288:z
5281:y
5274:x
5252:.
5248:]
5241:2
5236:1
5231:0
5226:0
5219:8
5214:0
5209:1
5204:0
5189:0
5184:0
5179:1
5172:[
5164:]
5157:2
5152:1
5147:0
5142:0
5135:8
5130:0
5125:1
5120:0
5113:9
5108:0
5103:3
5098:1
5091:[
5083:]
5076:2
5071:1
5066:0
5061:0
5054:8
5049:0
5044:1
5039:0
5032:5
5027:2
5019:3
5014:1
5007:[
4999:]
4992:2
4987:1
4982:0
4977:0
4970:2
4965:3
4957:1
4952:0
4945:5
4940:2
4932:3
4927:1
4920:[
4905:]
4898:2
4890:7
4885:2
4877:0
4870:2
4865:3
4857:1
4852:0
4845:5
4840:2
4832:3
4827:1
4820:[
4812:]
4805:2
4797:7
4792:2
4784:0
4777:8
4764:4
4756:0
4749:5
4744:2
4736:3
4731:1
4724:[
4716:]
4709:8
4704:3
4699:4
4694:2
4687:8
4674:4
4666:0
4659:5
4654:2
4646:3
4641:1
4634:[
4622:]
4615:8
4610:3
4605:4
4600:2
4593:7
4588:6
4583:5
4578:3
4571:5
4566:2
4558:3
4553:1
4546:[
4509:.
4468:.
4463:]
4456:8
4451:3
4446:4
4441:2
4434:7
4429:6
4424:5
4419:3
4412:5
4407:2
4399:3
4394:1
4387:[
4338:)
4335:2
4332:,
4329:8
4326:,
4317:(
4314:=
4311:)
4308:z
4305:,
4302:y
4299:,
4296:x
4293:(
4283:x
4279:z
4275:y
4271:y
4267:z
4246:2
4243:=
4240:z
4230:1
4227:+
4224:z
4218:2
4215:7
4209:=
4206:2
4203:+
4200:z
4197:3
4180:y
4156:1
4153:+
4150:z
4144:2
4141:7
4135:=
4132:y
4125:2
4122:+
4119:z
4116:3
4113:=
4110:y
4104:{
4079:y
4076:3
4070:z
4067:2
4064:+
4061:5
4058:=
4055:x
4045:x
4021:8
4018:=
4015:z
4012:3
4009:+
4006:y
4003:4
4000:+
3997:x
3994:2
3987:7
3984:=
3981:z
3978:6
3975:+
3972:y
3969:5
3966:+
3963:x
3960:3
3953:5
3950:=
3947:z
3944:2
3938:y
3935:3
3932:+
3929:x
3923:{
3883:z
3879:y
3875:x
3857:.
3851:7
3848:1
3840:x
3835:7
3832:1
3824:=
3821:z
3803:7
3795:+
3792:x
3787:7
3784:3
3776:=
3773:y
3757:z
3742:y
3738:x
3734:z
3730:z
3726:y
3722:x
3718:z
3700:.
3696:2
3693:+
3690:z
3687:3
3684:=
3681:y
3665:1
3659:z
3656:7
3650:=
3647:x
3616:7
3609:=
3602:z
3599:6
3592:+
3585:y
3582:5
3575:+
3568:x
3565:3
3557:5
3550:=
3543:z
3540:2
3526:y
3523:3
3516:+
3509:x
3460:)
3457:6
3453:,
3450:2
3443:,
3440:3
3437:(
3409:)
3406:6
3403:=
3400:z
3396:,
3393:2
3387:=
3384:y
3380:,
3377:3
3374:=
3371:x
3368:(
3315:k
3311:k
3278:y
3274:x
3272:3
3250:3
3243:=
3236:y
3233:2
3226:+
3219:x
3216:3
3208:1
3201:=
3194:y
3187:+
3180:x
3177:2
3169:1
3162:=
3155:y
3148:+
3141:x
3093:=
3090:y
3087:2
3084:+
3081:x
3078:3
3062:6
3059:=
3056:y
3053:2
3050:+
3047:x
3044:3
3006:y
3002:x
3000:3
2994:y
2990:x
2988:3
2947:7
2940:=
2933:y
2930:3
2923:+
2916:x
2913:4
2905:8
2898:=
2891:y
2888:5
2881:+
2874:x
2871:3
2863:1
2853:=
2846:y
2843:2
2829:x
2793:=
2790:y
2787:4
2784:+
2781:x
2778:6
2762:6
2759:=
2756:y
2753:2
2750:+
2747:x
2744:3
2725:y
2721:x
2719:4
2713:y
2709:x
2707:3
2701:y
2697:x
2595:m
2591:n
2586:m
2582:n
2571:.
2561:.
2534:n
2524:n
2515:n
2487:-
2477:y
2473:x
2462:.
2455:.
2448:.
2413:n
2409:x
2405:,
2399:,
2394:2
2390:x
2386:,
2381:1
2377:x
2354:y
2350:x
2348:3
2342:y
2338:x
2305:.
2300:]
2292:m
2288:b
2271:2
2267:b
2257:1
2253:b
2246:[
2241:=
2237:b
2232:,
2227:]
2219:n
2215:x
2198:2
2194:x
2184:1
2180:x
2173:[
2168:=
2164:x
2159:,
2154:]
2146:n
2143:m
2139:a
2126:2
2123:m
2119:a
2111:1
2108:m
2104:a
2072:n
2069:2
2065:a
2048:a
2036:a
2026:n
2023:1
2019:a
2002:a
1990:a
1983:[
1978:=
1975:A
1963:m
1959:b
1955:n
1947:x
1943:n
1941:×
1939:m
1935:A
1920:b
1916:=
1912:x
1908:A
1886:m
1882:n
1878:m
1825:]
1817:m
1813:b
1796:2
1792:b
1782:1
1778:b
1771:[
1766:=
1761:]
1753:n
1750:m
1746:a
1729:n
1726:2
1722:a
1712:n
1709:1
1705:a
1698:[
1691:n
1687:x
1683:+
1677:+
1672:]
1664:2
1661:m
1657:a
1636:a
1622:a
1615:[
1608:2
1604:x
1600:+
1595:]
1587:1
1584:m
1580:a
1559:a
1545:a
1538:[
1531:1
1527:x
1462:m
1458:b
1454:,
1448:,
1443:2
1439:b
1435:,
1430:1
1426:b
1403:n
1400:m
1396:a
1392:,
1386:,
1377:a
1373:,
1364:a
1341:n
1337:x
1333:,
1327:,
1322:2
1318:x
1314:,
1309:1
1305:x
1274:,
1269:m
1265:b
1261:=
1256:n
1252:x
1246:n
1243:m
1239:a
1235:+
1229:+
1224:2
1220:x
1214:2
1211:m
1207:a
1203:+
1198:1
1194:x
1188:1
1185:m
1181:a
1164:2
1160:b
1156:=
1151:n
1147:x
1141:n
1138:2
1134:a
1130:+
1124:+
1119:2
1115:x
1105:a
1101:+
1096:1
1092:x
1082:a
1072:1
1068:b
1064:=
1059:n
1055:x
1049:n
1046:1
1042:a
1038:+
1032:+
1027:2
1023:x
1013:a
1009:+
1004:1
1000:x
990:a
983:{
961:n
957:m
929:2
926:3
921:=
918:x
898:x
878:1
875:=
872:y
852:y
826:=
823:y
820:9
817:+
813:)
809:y
804:2
801:3
793:3
789:(
785:4
772:x
751:.
748:y
743:2
740:3
732:3
729:=
726:x
703:y
683:x
656:.
642:=
635:y
632:9
625:+
618:x
615:4
607:6
600:=
593:y
590:3
583:+
576:x
573:2
532:=
529:x
503:4
500:=
497:x
494:2
348:,
345:)
342:2
336:,
333:2
327:,
324:1
321:(
318:=
315:)
312:z
309:,
306:y
303:,
300:x
297:(
277:z
273:y
269:x
244:0
241:=
238:z
232:y
227:2
224:1
219:+
216:x
206:2
200:=
197:z
194:4
191:+
188:y
185:2
179:x
176:2
169:1
166:=
163:z
157:y
154:2
151:+
148:x
145:3
139:{
85:)
79:(
74:)
70:(
56:.
20:)
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