# Fundamental theorem of linear algebra

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

${\displaystyle A=U{\mathit {\Sigma }}V^{\mathrm {T} }}$

First, each matrix ${\displaystyle A\in \mathbb {R} ^{m\times n}}$ (${\displaystyle A}$ has ${\displaystyle m}$ rows and ${\displaystyle n}$ columns) induces four fundamental subspaces. These fundamental subspaces are as follows:

name of subspace definition containing space dimension basis
column space, range or image ${\displaystyle \operatorname {im} (A)}$ or ${\displaystyle \operatorname {range} (A)}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle r}$ (rank) The first ${\displaystyle r}$ columns of ${\displaystyle U}$
nullspace or kernel ${\displaystyle \ker(A)}$ or ${\displaystyle \operatorname {null} (A)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n-r}$ (nullity) The last ${\displaystyle (n-r)}$ columns of ${\displaystyle V}$
row space or coimage ${\displaystyle \operatorname {im} (A^{\mathrm {T} })}$ or ${\displaystyle \operatorname {range} (A^{\mathrm {T} })}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle r}$ (rank) The first ${\displaystyle r}$ columns of ${\displaystyle V}$
left nullspace or cokernel ${\displaystyle \ker(A^{\mathrm {T} })}$ or ${\displaystyle \operatorname {null} (A^{\mathrm {T} })}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle m-r}$ (corank) The last ${\displaystyle (m-r)}$ columns of ${\displaystyle U}$

Secondly:

1. In ${\displaystyle \mathbb {R} ^{n}}$, ${\displaystyle \ker(A)=(\operatorname {im} (A^{\mathrm {T} }))^{\perp }}$, that is, the nullspace is the orthogonal complement of the row space
2. In ${\displaystyle \mathbb {R} ^{m}}$, ${\displaystyle \ker(A^{\mathrm {T} })=(\operatorname {im} (A))^{\perp }}$, 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 ${\displaystyle A\colon V\to W}$ and ${\displaystyle A^{*}\colon W^{*}\to V^{*}}$: the kernel and image of ${\displaystyle A^{*}}$ are the cokernel and coimage of ${\displaystyle A}$.