Fundamental Matrix Subspaces


Given a real m×n matrix A, there are four associated vector subspaces which are known colloquially as its fundamental subspaces, namely the column spaces and the null spaces of the matrices A and its transpose A^(T). These four subspaces are important for a number of reasons, one of which is the crucial role they play in the so-called fundamental theorem of linear algebra.

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.

In the event that m=n=2, all four of the fundamental matrix subspaces are lines in R^2. In this case, one can write A=xy^(T) for some 2×1 vectors x,y, whereby the directions of the four lines correspond to x, x^_|_, y, and y^_|_. An elementary fact from linear algebra is that these directions are also represented by the eigenvectors of A and A^(T) (Strang 2008); this is one of the reasons why the four fundamental subspaces of A are often associated with the eigenvalues and the singular value decompositions of A and A^(T) in many presentations of the fundamental theorem of linear algebra (Strang 2012).

See also

Column Space, Eigenvector, Fundamental Theorem of Linear Algebra, Matrix, Null Space, Orthogonal Set, Range, Row Space, Singular Value Decomposition, Subspace, Vector Space

This entry contributed by Christopher Stover

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Strang, G. "The Fundamental Theorem of Linear Algebra." Amer. Math. Monthly 100, 848-855, 1993.Strang, G. "The Four Fundamental Subspaces: 4 Lines." 2008.

Cite this as:

Stover, Christopher. "Fundamental Matrix Subspaces." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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