Search results
Results from the WOW.Com Content Network
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [ 1 ] [ 2 ] [ 3 ] This corresponds to the maximal number of linearly independent columns of A .
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 ).
Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall 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
Linear algebra is the branch of mathematics concerning linear equations such as: ... which include the computation of the ranks, kernels, matrix inverses. ...
As a result, it makes sense to define the k-rank or split rank of a group G over k as the dimension of any maximal split torus in G over k. For any maximal torus T in a linear algebraic group G over a field k , Grothendieck showed that T k ¯ {\displaystyle T_{\overline {k}}} is a maximal torus in G k ¯ {\displaystyle G_{\overline {k}}} . [ 12 ]
There are several equivalent definitions, all modifying the definition of the linear rank slightly. Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative m×n-matrix A is equal to the smallest number q such there exists a nonnegative m×q-matrix B and a nonnegative q×n-matrix C such that A = BC (the usual matrix product).
Every finite-dimensional matrix has a rank decomposition: ... (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed ...
Applicable to: m-by-n matrix A of rank r Decomposition: A = C F {\displaystyle A=CF} where C is an m -by- r full column rank matrix and F is an r -by- n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A , [ 2 ] which one can apply to obtain all solutions of the linear system A x ...