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

Secondly:

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