The four subspaces

Row, null, column and left null spaces

This chapter develops the four fundamental subspaces associated with a matrix A. It connects the algebraic definitions to geometric intuition, then relates the picture to Aᵀ and AᵀA.

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Chapter contents

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Topic	Pages
Definitions	1
The four subspaces of A: introduction	2–4
Geometric interpretation of the four subspaces of A	5–6
The four subspaces of A: summary table	7–8
Geometric view of transformations by A, Aᵀ & AᵀA	9–10
The four subspaces of Aᵀ	11
The four subspaces of AᵀA	12–13
Why do we need this abstract concept?	14–16
Tying it all together	17–18

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