16.18 Tensor Alternating Transformation Behavior

Tensor Alternating Transformation Behavior is the overarching description of how alternating tensors and exterior powers respond to linear maps, encompassing the preservation of alternation under pullback, the induced action on exterior powers, the scaling behavior at the top degree governed by the determinant, and the functorial compatibility of these operations with composition of linear maps. It unifies the various individual transformation properties of alternating tensors into a single coherent account of how this entire algebraic structure interacts with linear change.

The Two Faces of Transformation

Pullback on Alternating Forms

Given a linear map T from V to W and an alternating k-linear form ω on W, the pullback T*ω is defined on V by precomposing with T in every argument. Alternating transformation behavior guarantees that this pullback remains alternating on V, regardless of the properties of T, since repeated arguments in V map to repeated arguments in W, and ω's vanishing on repetition transfers directly.

Pushforward on Exterior Powers

Dually, T induces a map Λᵏ(T) on exterior powers, sending simple wedge products v₁ ∧ ... ∧ vₖ to their images Tv₁ ∧ ... ∧ Tvₖ. This assignment extends uniquely to all of Λᵏ(V) by linearity, and the well-definedness of this extension is itself a manifestation of alternating transformation behavior, since it depends on the antisymmetry relations of Λᵏ(V) being compatible with the linearity of T.

Degree-Specific Behavior

General Degree Compatibility

At an arbitrary degree k, the induced map Λᵏ(T) acts on the C(n, k)-dimensional space Λᵏ(V) (for T mapping an n-dimensional space to itself) by some linear transformation whose matrix entries are determined by minors of the matrix representing T, reflecting how T mixes together different k-element combinations of basis directions.

Top Degree Scaling by the Determinant

At the top degree k = n, where Λⁿ(V) is one-dimensional, the induced map Λⁿ(T) reduces to multiplication by a single scalar, which is precisely det(T):

for any ω in Λⁿ(V). This scaling behavior is the most consequential special case of alternating transformation behavior, directly connecting the abstract theory of exterior powers to determinant computation and volume scaling.

Functorial Consistency

Compatibility With Composition

For linear maps S and T, alternating transformation behavior guarantees:

meaning the exterior power construction respects composition of linear maps at every degree, not merely at the top degree. This is what makes the exterior power operation functorial, transforming linear algebra problems about V and W into corresponding problems about Λᵏ(V) and Λᵏ(W) in a structurally consistent way.

Identity and Invertibility Preservation

Applying an identity map induces the identity map on every exterior power, and if T is invertible, Λᵏ(T) is invertible as well, with inverse Λᵏ(T⁻¹). This preservation of invertibility, combined with the determinant scaling relation, explains why T is invertible exactly when det(T) is nonzero: noninvertibility of T at the vector space level corresponds precisely to the collapse of the top exterior power map to the zero map.

Behavior Under Direct Sums and Restriction

Compatibility With Subspace Restriction

If T maps a subspace U of V into itself, the induced exterior power map Λᵏ(T) restricts consistently to Λᵏ(U) as a subspace of Λᵏ(V), meaning alternating transformation behavior respects the inclusion of invariant subspaces, an important consideration when analyzing block-triangular or invariant-subspace decompositions of linear maps.

Interaction With Direct Sum Decompositions

When V decomposes as a direct sum of T-invariant subspaces, the exterior power Λᵏ(V) decomposes correspondingly into pieces built from wedge products drawn across the summands, and the induced map Λᵏ(T) respects this decomposition, acting on each piece according to the restricted behavior of T on the relevant summands.

Significance of the Behavior

Alternating transformation behavior is the comprehensive account of how linear maps interact with the entire graded structure of exterior powers, from arbitrary intermediate degrees down to the determinant-scaling top degree. It guarantees functorial consistency under composition, ties invertibility of a map directly to the nonvanishing of its determinant, and provides the structural foundation for pullback and pushforward operations used throughout differential geometry, multilinear algebra, and the theory of determinants.