Fundamental Theorem of Linear Algebra

Given an m×n matrix A, the fundamental theorem of linear algebra is a collection of results relating various properties of the four fundamental matrix subspaces of A. In particular:

1. dimR(A)=dimR(A^(T)) and dimR(A)+dimN(A)=n where here, R(A) denotes the range or column space of A, A^(T) denotes its transpose, and N(A) denotes its null space.

2. The null space N(A) is orthogonal to the row space R(A^(T)).

1. There exist orthonormal bases for both the column space R(A) and the row space R(A^(T)) of A.

4. With respect to the orthonormal bases of R(A) and R(A^(T)), A is diagonal.

The third item on this list stems from Gram-Schmidt Orthonormalization; the fourth item stems from the singular value decomposition of A. Also, while different, the first item is reminiscent of the rank-nullity theorem.


The above figure summarizes some of the interactions between the four fundamental matrix subspaces for a real m×n matrix A including whether the spaces in question are subspaces of R^m or R^n, which subspaces are orthogonal to one another, and how the matrix A maps various vectors x relative to the subspace in which x lies. This diagram essentially makes visual the first two parts of the above-stated result.

Worth noting is that this theorem is often stated differently and with varying numbers of parts. In particular, it is relatively common for versions of this theorem to include only the first two items given above, though the importance of the last two items is often cited to justify stating a four-part version like the above (Strang 1993). Some authors also include corollaries of the above statements within the statements themselves (Badger 2012).

See also

Column Space, Eigenvector, Fundamental Matrix Subspaces, Gram-Schmidt Orthonormalization, Matrix, Null Space, Orthogonal Set, Orthonormal Basis, Range, Rank-Nullity Theorem, Row Space, Singular Value Decomposition, Subspace, Vector Space

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha


Badger, M. "Fundamental Theorem of Linear Algebra." 2012., G. "The Fundamental Theorem of Linear Algebra." Amer. Math. Monthly 100, 848-855, 1993.

Cite this as:

Stover, Christopher. "Fundamental Theorem of Linear Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications