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For example, given a linear map T : V → W, the image T(V) of V, and the inverse image T −1 (0) of 0 (called kernel or null space), are linear subspaces of W and V, respectively. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums
Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7 Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra , Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8 , archived from the original on March 1, 2001
Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. [3] [4] Computing matrix products is a central operation in all computational applications of linear algebra.
The equation + = is a linear system with unknowns and the same number of equations. Hence it is uniquely solvable for any given C {\displaystyle C} if and only if the homogeneous equation A X + X B = 0 {\displaystyle AX+XB=0} admits only the trivial solution 0 {\displaystyle 0} .
Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices. If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication =, where the eigenvector v ...
The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M ; and the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f ) and the nullity of f (the dimension of the kernel of f ).
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