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

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Adjoints satisfy $(S + T)^{} = S^{} + T^{}$ , $(λ T)^{} = \overline{λ} T^{}$ , $(S T)^{} = T^{} S^{}$ , and $T^{* *} = T$

Statement

For compatible linear maps between finite-dimensional real or complex inner product spaces,

$(S + T)^{*} = S^{*} + T^{*}, (λ T)^{*} = \overline{λ} T^{*}, (S T)^{*} = T^{*} S^{*}, T^{* *} = T .$

Also $I^{*} = I$ and $0^{*} = 0$ .

Facts & Assumptions

Given: Compatible finite-dimensional linear maps $S, T$ and a scalar $λ$ .

[L1]

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

[L2]

Finite-dimensional adjoints exist and are unique (Every linear map between finite-dimensional inner product spaces has a unique adjoint).

[L3]

Equality of pairings with every vector forces equality of the paired vectors (Inner products separate vectors, and the induced norm is homogeneous: $∥ λ v ∥ = ∣ λ ∣ ∥ v ∥$ ).

Proof

technique · direct

1.1

For all $v, w$ , linearity in the first argument and [L1] give $⟨ (S + T) v, w ⟩ = ⟨ v, (S^{*} + T^{*}) w ⟩$ . Uniqueness in [L2] proves the sum formula. The same calculation gives $I^{*} = I$ and $0^{*} = 0$ .

L1L2

1.2

Likewise $⟨ λ T v, w ⟩ = λ ⟨ v, T^{*} w ⟩ = ⟨ v, \overline{λ} T^{*} w ⟩$ , because the second argument is conjugate-linear. Thus $(λ T)^{*} = \overline{λ} T^{*}$ .

L1L2algebra

1.3

For a composite, $⟨ S T v, w ⟩ = ⟨ T v, S^{*} w ⟩ = ⟨ v, T^{*} S^{*} w ⟩$ , so uniqueness gives $(S T)^{*} = T^{*} S^{*}$ .

L1L2

2.1

Conjugate symmetry rewrites [L1] as $⟨ T^{*} w, v ⟩ = ⟨ w, T v ⟩$ , so $T$ is an adjoint of $T^{*}$ . By [L2], $T^{* *} = T$ .

L1L2L3∎

Depends on

Used by

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Direct dependencies and their dependencies through the next three levels: 36 results over 12 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

Sheldon Axler, Linear Algebra Done Right, 4th ed., result 7.5 (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, §5.5.1, useful formulas for adjoints (standard reference, not scraped)

Adjoints satisfy (S+T)∗=S∗+T∗(S+T)∗=S∗+T∗, (λT)∗=λ‾T∗(λT)∗=λT∗, (ST)∗=T∗S∗(ST)∗=T∗S∗, and T∗∗=TT∗∗=T