PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Linear pullback respects tensor products and permutations

Statement

Let $V$ and $W$ be finite-dimensional real vector spaces, let $A : V \to W$ be linear, let $S$ and $T$ be covariant tensors on $W$ of degrees $k$ and $ℓ$ , respectively, and let $σ \in S_{ℓ}$ . Then

$A^{*} (S \otimes T) = A^{*} S \otimes A^{*} T, A^{*} (σ \cdot T) = σ \cdot (A^{*} T) .$

Facts & Assumptions

Given: Finite-dimensional real vector spaces $V, W$ , a linear map $A : V \to W$ , covariant tensors $S, T$ on $W$ of degrees $k, ℓ$ , and a permutation $σ \in S_{ℓ}$ .

[F1]

Pullback of a covariant tensor substitutes $A v_{i}$ into every slot (The pullback of a covariant tensor by a linear map).

[F2]

Tensor product multiplies the factor values on concatenated arguments, and the permutation action reorders the arguments (The tensor product of multilinear tensors, The permutation action on covariant tensors).

Proof

technique · direct

1.1F1F2givenalgebra

Evaluating on $v_{1}, \dots, v_{k + ℓ}$ and using [F1] and [F2], $A^{*} (S \otimes T) (v_{1}, \dots, v_{k + ℓ}) = (S \otimes T) (A v_{1}, \dots, A v_{k + ℓ}) = S (A v_{1}, \dots, A v_{k}) T (A v_{k + 1}, \dots, A v_{k + ℓ}),$ which is exactly $(A^{*} S \otimes A^{*} T) (v_{1}, \dots, v_{k + ℓ})$ .

1.2F1F2givenalgebra

Likewise, $A^{*} (σ \cdot T) (v_{1}, \dots, v_{ℓ}) = (σ \cdot T) (A v_{1}, \dots, A v_{ℓ}) = T (A v_{σ (1)}, \dots, A v_{σ (ℓ)}),$ which equals $(σ \cdot A^{*} T) (v_{1}, \dots, v_{ℓ})$ .

2.1step 1.1step 1.2∎

Therefore linear pullback respects tensor products and permutations.

Depends on

Used by

Pullback of covariant tensors is smooth and functorial Proposition
Pullback of forms is smooth functorial and preserves wedges Proposition

Dependency tree · two levels

5 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)