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 singular value decomposition:

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

First, each matrix $A \in \mathbf{R}^{m \times n}$ ($A$ 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	$\mathrm{im}(A)$ or $\mathrm{range} (A)$	$\mathbf{R}^m$	$r$ (rank)	The first $r$ columns of $U$
nullspace or kernel	$\mathrm{ker}(A)$ or $\mathrm{null} (A)$	$\mathbf{R}^n$	$n - r$ (nullity)	The last $(n - r)$ columns of $V$
row space or coimage	$\mathrm{im}(A^\mathrm{T})$ or $\mathrm{range} (A^\mathrm{T})$	$\mathbf{R}^n$	$r$ (rank)	The first $r$ columns of $V$
left nullspace or cokernel	$\mathrm{ker}(A^\mathrm{T})$ or $\mathrm{null} (A^\mathrm{T})$	$\mathbf{R}^m$	$m - r$ (corank)	The last $(m - r)$ columns of $U$

Secondly:

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

See also

• Rank–nullity theorem
• Closed range theorem

References

• Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988.
• Strang, Gilbert (1993), "The fundamental theorem of linear algebra" (PDF), American Mathematical Monthly 100 (9): 848–855, doi:10.2307/2324660, JSTOR 2324660
• Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388

External links

• Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces at Google Video, from MIT OpenCourseWare