2.22.4 Tensor Linear Map Scalar Preservation

Tensor Linear Map Scalar Preservation is the property that the tensor identification of linear maps as elements of V* ⊗ W respects scalar multiplication on both sides of the correspondence, so that the tensor representing a scalar multiple of a linear map equals that same scalar multiple of the tensor representing the original map, and this holds regardless of whether the scalar is absorbed into the map itself, into a chosen dual-vector factor, or into a chosen vector factor of a simple tensor. Together with additivity preservation, scalar preservation completes the demonstration that Hom(V, W) and V* ⊗ W are isomorphic as vector spaces, not merely in bijection as sets.

Scalar Multiplication on the Side of Linear Maps

Pointwise Scalar Multiple Definition

For φ ∈ Hom(V, W) and a scalar c ∈ F, the scalar multiple cφ is defined pointwise by:

for every v ∈ V, matching the ordinary definition of scalar multiplication of functions.

Scalar Preservation Under the Correspondence

If φ corresponds to the tensor α ∈ V* ⊗ W, scalar preservation states that cφ corresponds to cα, the ordinary scalar multiple of α as an element of the vector space V* ⊗ W. No rescaling of individual tensor factors is required to state this correctly; the claim is about the tensor α as a whole being scaled by c.

Where the Scalar May Be Absorbed

Absorption Into Either Factor of a Simple Tensor

For a simple tensor ω ⊗ w, the defining bilinearity of the tensor product gives:

so the scalar c can equivalently be absorbed into the dual-vector factor, into the vector factor, or left outside the tensor product entirely, and all three expressions denote the same element of V* ⊗ W. This flexibility mirrors the corresponding freedom on the side of linear maps: (cφ)(v) = c(ω(v)) w = (cω)(v) w = ω(v)(cw) all describe the same rank-one map.

No Ambiguity for General Tensors

For a general element α = Σ ω_k ⊗ w_k of V* ⊗ W, written as a sum of simple tensors, cα = Σ (c ω_k) ⊗ w_k = Σ ω_k ⊗ (c w_k), and any mixture of these placements across different terms of the sum, gives the identical result cα; scalar preservation holds regardless of how the scalar is distributed among the terms, since the underlying tensor is determined by the sum as a whole.

Verification in Coordinates

Matrix Scalar Multiplication Matches Tensor Scalar Multiplication

Relative to bases {e_i} of V and {f_j} of W, if φ has matrix entries a^j_i, then cφ has matrix entries c · a^j_i, since (cφ)(e_i) = c φ(e_i) = c Σ_j a^j_i f_j = Σ_j (c a^j_i) f_j. This matches exactly the entrywise scalar multiple of the tensor Σ a^j_i e^i ⊗ f_j, confirming in coordinates that scalar multiplication of matrices is the coordinate expression of scalar multiplication of the corresponding (1, 1)-tensors.

Scalar Preservation as Part of Linearity of the Identification

Together With Additivity, Establishing a Vector-Space Isomorphism

The identification map Θ : V* ⊗ W → Hom(V, W), sending ω ⊗ w to the rank-one map v ↦ ω(v) w and extended linearly, satisfies Θ(cα) = cΘ(α) precisely because scalar preservation holds; combined with additivity preservation, Θ(α + β) = Θ(α) + Θ(β), this shows Θ is a linear map. Since Θ is also bijective, matching bases as described in tensor linear map structure, Θ is a full vector-space isomorphism, and scalar preservation is the second of the two properties, alongside additivity, that this isomorphism claim requires.

Consistency With Scalar Multiplication by Zero and One

Setting c = 1 gives 1 · φ = φ and 1 · α = α, consistent trivially. Setting c = 0 gives the zero map and the zero tensor respectively, matching the earlier observation that Θ sends the zero tensor to the zero map; scalar preservation at these boundary values confirms that the correspondence behaves correctly at the extremes of scalar multiplication, not merely for generic nonzero scalars.