The four fundamental subspaces

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.

• In the domain, every x decomposes uniquely into a row-space component and a null-space component
• In the codomain, every b decomposes uniquely into a column-space component and a left-null-space component
• row(A) ⟂ null(A) and col(A) ⟂ ℓ-null(A)
• dim(row(A)) = dim(col(A)) = rank(A)
• null(A) describes inputs sent to 0, while ℓ-null(A) describes output directions that A cannot produce

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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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.

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