Linear Algebra Done Right · Problem 9D.7

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · tensor-products

Problem

Suppose dim⁡V>2\dim V > 2 and dim⁡W>2\dim W > 2. Prove that {v1⊗w1+v2⊗w2:v1,v2∈V and w1,w2∈W}≠V⊗W.\{v_1 \otimes w_1 + v_2 \otimes w_2 : v_1, v_2 \in V \text{ and } w_1, w_2 \in W\} \neq V \otimes W.

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