TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

For a linear map $T : V \to W$ between finite-dimensional inner-product spaces, $x$ minimises $∥ T x - b ∥$ if and only if $T^{} (T x - b) = 0$ , equivalently $T^{} T x = T^{*} b$ ; minimisers exist and any two differ by an element of $\ker T$

Statement

Let $T : V \to W$ be a linear map between finite-dimensional inner product spaces and let $b \in W$ . A vector $x \in V$ minimises $∥ T x - b ∥$ if and only if

$T^{*} (T x - b) = 0,$

equivalently $T^{*} T x = T^{*} b$ . Minimisers exist, and if $x_{0}$ is one minimiser, then the full set of minimisers is $x_{0} + \ker T$ .

Facts & Assumptions

Given: A finite-dimensional map $T : V \to W$ and $b \in W$ .

[L1]

Orthogonal projection onto a finite-dimensional subspace is its unique nearest point (The orthogonal projection is the unique nearest point in the subspace).

[L2]

The adjoint is defined by $⟨ T v, w ⟩ = ⟨ v, T^{*} w ⟩$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ ).

[L3]

The identity $\ker T^{*} = (im T)^{⊥}$ holds in finite dimension ( $\ker T^{*} = (im T)^{⊥}$ and $im T^{*} = (\ker T)^{⊥}$ in finite dimension).

Proof

technique · direct

1.1

The subspace $im T$ has the unique nearest point $P_{im T} b$ to $b$ by [L1]. Choose $x_{0}$ with $T x_{0} = P_{im T} b$ . Thus a minimiser exists.

L1choose

2.1

A vector $x$ is a minimiser exactly when $T x = P_{im T} b$ , which by orthogonal projection is exactly when $b - T x \in (im T)^{⊥}$ . By [L3], this is exactly $T^{*} (b - T x) = 0$ .

step 1.1L1L3

3.1

Linearity turns the last equation into $T^{*} T x = T^{*} b$ , equivalently $T^{*} (T x - b) = 0$ .

step 2.1L2algebra

4.1

If $x$ and $x_{0}$ are minimisers, uniqueness of the nearest image point gives $T x = T x_{0}$ , so $x - x_{0} \in \ker T$ . Conversely, adding any element of $\ker T$ leaves the image and residual unchanged. Hence the minimisers are exactly $x_{0} + \ker T$ .

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Sergei Treil, Linear Algebra Done Wrong, §5.4.1 (standard reference, not scraped)

For a linear map T:V→WT:V→W between finite-dimensional inner-product spaces, xx minimises ∥Tx−b∥∥Tx−b∥ if and only if T∗(Tx−b)=0T∗(Tx−b)=0, equivalently T∗Tx=T∗bT∗Tx=T∗b; minimisers exist and any two differ by an element of ker⁡TkerT