LA-I's four fundamental subspaces gave every matrix four attached subspaces, with dimensions that fit together (, , , ) but with no particular relationship between their bases — the column space and row space, for instance, were described by whatever basis elimination happened to produce. This note shows the SVD does something LA-I's elimination-based construction never could: it hands all four subspaces orthonormal bases simultaneously, drawn from the same two orthogonal matrices , with the split between subspaces falling exactly at the rank .
Learning objectives. By the end of this note you should be able to:
- identify which columns of and span each of the four fundamental subspaces;
- explain why the SVD's split is exactly the rank ;
- verify the orthogonality between row space/nullspace and column space/left nullspace using the SVD directly;
- state why this is a strictly stronger result than LA-I's elimination-based construction.
In this lesson
- The four subspaces, read off the SVD
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