TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

1.1L1choose

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.

2.1step 1.1L1L3

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

3.1step 2.1L2algebra

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

4.1step 1.1step 2.1∎

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 $x_{0} + \ker T$ .

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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