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"×"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 or image	$mathrm\{im\}(A)$	$mathbf\{R\}^m$	$r$	The $r$ columns corresponding to those with pivots in $mathbf\{U\}$
nullspace or kernel	$mathrm\{ker\}(A)$	$mathbf\{R\}^n$	$n\; -\; r$ (nullity)	The $(n\; -\; r)$ columns of $x$ in the solution of $mathbf\{U\}mathbf\{x\}\; =\; mathbf\{0\}$
row space or coimage	$mathrm\{im\}(A^T)$	$mathbf\{R\}^n$	$r$	The $r$ rows corresponding to those with pivots in $mathbf\{U\}$
left nullspace or cokernel	$mathrm\{ker\}(A^T)$	$mathbf\{R\}^m$	$m\; -\; r$	The last $(m\; -\; r)$ rows of $mathbf\{L\}^\{-1\}mathbf\{P\}$

Secondly:

# In Rⁿ: $mathrm\{ker\}(A)\; =\; (mathrm\{im\}(A^T))^perp$, 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

References

* Strang, Gilbert. "Linear Algebra and Its Applications". 3rd ed. Orlando: Saunders, 1988.