Fundamental theorem of linear algebra
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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:
First, each matrix (
has
rows and
columns) induces four fundamental subspaces. These fundamental subspaces are:
name of subspace | definition | containing space | dimension | basis |
---|---|---|---|---|
column space, range or image | ![]() ![]() |
![]() |
![]() |
The first ![]() ![]() |
nullspace or kernel | ![]() ![]() |
![]() |
![]() |
The last ![]() ![]() |
row space or coimage | ![]() ![]() |
![]() |
![]() |
The first ![]() ![]() |
left nullspace or cokernel | ![]() ![]() |
![]() |
![]() |
The last ![]() ![]() |
Secondly:
- In
,
, that is, the nullspace is the orthogonal complement of the row space
- In
,
, that is, the left nullspace is the orthogonal complement of the column space.
File:The four subspaces.svg
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 and
: the kernel and image of
are the cokernel and coimage of
.
See also
References
- Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988.
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External links
- Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces at Google Video, from MIT OpenCourseWare