Fundamental theorem of linear algebra

In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m×nmatrix A and its singular value decomposition:

First, each matrix  (A has m rows and n columns) induces four fundamental subspaces. These fundamental subspaces are:

name of subspace	definition	containing space	dimension	basis
column space, range or image	im(A) or range(A)		r (rank)	The first r columns of
nullspace or kernel	ker(A) or null(A)		n − r (nullity)	The last (n − r) columns of
row space or coimage	im(AT) or range(AT)		r	The first r rows of
left nullspace or cokernel	ker(AT) or null(AT)		m − r	The last (m − r) rows of