Every matrix A has two complementary subspaces in the domain, row(A) and null(A), and two complementary subspaces in the codomain, col(A) and ℓ-null(A). These pairs are orthogonal, so ℝⁿ = row(A) ⊕ null(A) and ℝᵐ = col(A) ⊕ ℓ-null(A).
The four subspaces organize what a linear map can do to inputs and outputs. They separate the directions that survive the transformation from the directions that disappear or cannot be reached.
This chapter provides multiple visual viewpoints—domain, codomain and geometric—to make the structure of the four subspaces clear and intuitive.
This framework explains least squares as projection and connects rank, one-to-one and onto through the structure of the four subspaces.
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| Topic | Pages |
|---|---|
| List & dimensions | 1–4 |
| Four subspaces, codomain view | 5–6 |
| Four subspaces, domain view | 7–8 |
| The four subspaces of A: summary table | 9–12 |
| Geometric view of transformations by A, Aᵀ & AᵀA | 13–15 |
| The four subspaces of Aᵀ | 16–17 |
| The four subspaces of AᵀA | 17–19 |
| Why do we need this abstract concept? | 19–21 |
| Tying it all together | 22–23 |
null(A) consists of input directions that A collapses to 0, while ℓ-null(A) consists of output directions orthogonal to col(A). So null(A) lives in the domain and measures failure of one-to-one, while ℓ-null(A) lives in the codomain and measures failure of onto.