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:

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

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

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

Secondly:

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

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 $A$ .

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, CiteSeerX 10.1.1.384.2309, 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

Media related to Fundamental theorem of linear algebra at Wikimedia Commons
Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces on YouTube, from MIT OpenCourseWare

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