Linear Algebra Done Right · Problem 9D.9

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · tensor-products

Problem

Suppose

S \in \mathcal{L}(V)

and

T \in \mathcal{L}(W)

. Prove that there exists a unique operator on

V \otimes W

that takes

v \otimes w

Sv \otimes Tw

for all

v \in V

and

w \in W

. In an abuse of notation, the operator on

V \otimes W

given by this exercise is often called

S \otimes T