Fundamental theorem of linear algebra

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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×n matrix A and its LDU factorization:

$PA=LDU\$

wherein P is a permutation matrix, L is a lower triangular matrix, D is a diagonal matrix, and U is an upper triangular matrix. At a more abstract level there is an interpretation that reads it in terms of a linear mapping and its transpose.

First, each matrix A induces four fundamental subspaces. These fundamental subspaces are:

name of subspace	definition	containing space	dimension	basis
column space, range or image	$im(A)$ or $range(A)$	$\mathbf{R}^m$	$r$ (rank)	The $r$ columns corresponding to those with pivots in $\mathbf{U}$
nullspace or kernel	$ker(A)$ or $null(A)$	$\mathbf{R}^n$	$n - r$ (nullity)	The $(n - r)$ columns of $x$ in the solution of $\mathbf{U}\mathbf{x} = \mathbf{0}$
row space or coimage	$im(A T)$ or $range(A T)$	$\mathbf{R}^n$	$r$	The $r$ rows corresponding to those with pivots in $\mathbf{U}$
left nullspace or cokernel	$ker(A T)$ or $null(A T)$	$\mathbf{R}^m$	$m - r$	The last $(m - r)$ rows of $\mathbf{L}^{-1}\mathbf{P}$

Secondly:

In $\mathbf{R}^n$ , $\mathrm{ker}(A) = (\mathrm{im}(A^T))^\perp$ , that is, the nullspace is the orthogonal complement of the row space
In $\mathbf{R}^m$ , $\mathrm{ker}(A^T) = (\mathrm{im}(A))^\perp$ , that is, the left nullspace is the orthogonal complement of the column space.

The four subspaces associated to a matrix A.

The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.

Further, all these spaces are intrinsically defined – they do not require a choice of basis – in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as $A\colon V \to W$ and $A^* \colon W^* \to V^*$ : the kernel and image of $A *$ are the cokernel and coimage of $A$ .