CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The rank of a linear map is the number of its nonzero singular values

Statement

Let $T : V \to W$ be a linear map between finite-dimensional real or complex inner product spaces. Then $rank T$ equals the number of positive singular values of $T$ .

Facts & Assumptions

Given: A linear map $T : V \to W$ between finite-dimensional real or complex inner product spaces, and an SVD $T v = \sum_{j = 1}^{r} s_{j} ⟨ v, e_{j} ⟩ f_{j}$ with $s_{1} \geq \dots \geq s_{r} > 0 = s_{r + 1} = \dots$ .

[L1]

Every linear map admits a singular value decomposition (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[L2]

Rank-nullity holds for linear maps with finite-dimensional domain (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

Proof

technique · direct

1.1

By [L1], one has $T v = 0$ exactly when $⟨ v, e_{j} ⟩ = 0$ for $1 \leq j \leq r$ , so $\ker T = span (e_{r + 1}, \dots, e_{n})$ and therefore $nullity T = n - r$ .

L1algebra

2.1

By [L2], $rank T + nullity T = n$ . Combining this with step 1.1 gives $rank T = r$ , the number of positive singular values.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)