Linear Algebra Done Right · Problem 9D.5

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · tensor-products

Problem

Suppose

m

and

n

are positive integers. For

v \in \mathbf{F}^m

and

w \in \mathbf{F}^n

, identify

v \otimes w

with an

m

-by-

n

matrix as in Example 9.76. With that identification, show that the set

\{v \otimes w : v \in \mathbf{F}^m \text{ and } w \in \mathbf{F}^n\}

is the set of

m

-by-

n

matrices (with entries in

\mathbf{F}

) that have rank at most one.