The Ultimate Fundamental theorem of linear algebra - American History Information Guide and Reference

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 $m \times n$ matrix $\mathbf{A}$ and its triangular or reduced factorization:

$\mathbf{P}\mathbf{A} = \mathbf{L}\mathbf{D}\mathbf{U}$ ,

wherein $\mathbf{P}$ is a permutation matrix, $\mathbf{L}$ is a lower triangular matrix, $\mathbf{D}$ is a diagonal matrix, and $\mathbf{U}$ 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 vector space $\mathbf{A}$ has four fundamental subspaces. These fundamental subspaces are:

name of subspace	containing space	Hamel dimension	relationship between $\mathbf{A}$ and $\mathbf{U}$	basis
column space or range	$\mathbf{R}^m$	$r$	$\mathfrak{R}(\mathbf{A}) \ne \mathfrak{R}(\mathbf{U})$	The $r$ columns corresponding to those with pivots in $\mathbf{U}$
nullspace or kernel	$\mathbf{R}^n$	$n - r$ (nullity)	$\mathfrak{N}(\mathbf{A}) = \mathfrak{N}(\mathbf{U})$	The $(n - r)$ columns of $x$ in the solution of $\mathbf{U}\mathbf{x} = \mathbf{0}$
row space	$\mathbf{R}^n$	$r$	$\mathfrak{R}(\mathbf{A}) = \mathfrak{R}(\mathbf{U})$	The $r$ rows corresponding to those with pivots in $\mathbf{U}$
left nullspace	$\mathbf{R}^m$	$m - r$	$\mathfrak{N}(\mathbf{A}) \ne \mathfrak{N}(\mathbf{U})$	The last $(m - r)$ rows of $\mathbf{L}^{-1}\mathbf{P}$

Secondly:

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

Reference

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