5.2.5 Tensor Product Map Area

The Tensor Product Map Area defines how tensor products create mappings between tensor spaces, central to multilinear algebra and algebraic structures.

Tensor Product Map Area is the detailed treatment of the induced linear map f ⊗ g built from linear maps f: V → V' and g: W → W', covering its construction via the universal property, the proof of its functorial identities, and the behavior of partial applications that act on only one tensor factor.

Constructing the Induced Map

From a Pair of Linear Maps to a Bilinear Map

Given f: V → V' and g: W → W', define

β : V \times W \to V^{'} \otimes W^{'}, β (v, w) = f (v) \otimes g (w)

This β is bilinear because f and g are linear and ⊗ itself is bilinear, so β inherits additivity and homogeneity in each argument from the composition of these already-bilinear and linear pieces.

Applying the Universal Property

Since β is bilinear, the universal property of V ⊗ W supplies a unique linear map f ⊗ g: V ⊗ W → V' ⊗ W' with (f ⊗ g)(v ⊗ w) = β(v, w) = f(v) ⊗ g(w). This is the definition of the induced map; its existence and uniqueness are inherited directly from the universal property area rather than requiring separate justification.

Functorial Identities

Compatibility with Composition

For f_1: V' → V'', f_2: V → V', g_1: W' → W'', g_2: W → W', both (f_1 ∘ f_2) ⊗ (g_1 ∘ g_2) and (f_1 ⊗ g_1) ∘ (f_2 ⊗ g_2) are linear maps V ⊗ W → V'' ⊗ W'', and they agree on every decomposable element:

((f_{1} \otimes g_{1}) \circ (f_{2} \otimes g_{2})) (v \otimes w) = f_{1} (f_{2}, (v)) \otimes g_{1} (g_{2}, (w))

which equals ((f_1 ∘ f_2) ⊗ (g_1 ∘ g_2))(v ⊗ w). Two linear maps agreeing on all decomposable elements agree everywhere, by the same spanning argument used to prove uniqueness in the universal property, so the two composites are equal.

Compatibility with Identity

Because id_V(v) ⊗ id_W(w) = v ⊗ w for all decomposable elements, the induced map id_V ⊗ id_W agrees with id_{V ⊗ W} on a spanning set, and therefore equals it. Together with compatibility under composition, this makes (V, W) ↦ V ⊗ W and (f, g) ↦ f ⊗ g behave as a functor of two linear-map arguments.

Partial Application to a Single Factor

Acting on One Tensor Slot

Setting g = id_W gives the map f ⊗ id_W: V ⊗ W → V' ⊗ W, sending v ⊗ w to f(v) ⊗ w and extending linearly; this map alters only the V-factor of every decomposable element while leaving every W-factor exactly as it was.

Injectivity and Surjectivity Transfer Componentwise

If f is injective, f ⊗ id_W is injective, and if f is surjective, f ⊗ id_W is surjective; both facts follow from choosing bases of V and W, expressing f ⊗ id_W in the induced basis {e_i ⊗ f_j}, and observing that the resulting block structure of the matrix mirrors the structure of f alone, factor by factor, block by block.