Linear Algebra Done Right · Problem 9D.8

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · tensor-products

Problem

Suppose

v_1, \ldots, v_m \in V

and

w_1, \ldots, w_m \in W

are such that

v_1 \otimes w_1 + \cdots + v_m \otimes w_m = 0.

Suppose that

U

is a vector space and

\Gamma\colon V \times W \to U

is a bilinear map. Show that

\Gamma(v_1, w_1) + \cdots + \Gamma(v_m, w_m) = 0.