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 \(\mathrm{range} (A) \)	\( \mathbf{R}^m \)	\( \) (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:

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
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 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

External links

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

Mathematics Encyclopedia