A First Course in Linear Algebra · Problem M60

Representations — Robert Beezer

Mathematics — Linear Algebra · zero-linear-transformation-invertibility

Problem

Suppose

U

and

V

are vector spaces. Define the function

Z: U \to V

Z(\mathbf{u}) = \mathbf{0}_V

for every

\mathbf{u} \in U

. Then by Exercise LT.M60,

Z

is a linear transformation. Formulate a condition on

U

and

V

that is equivalent to

Z

being an invertible linear transformation. In other words, fill in the blank to complete the following statement (and then give a proof):

Z

is invertible if and only if

U

and

V

are \_\_\_\_\_\_\_\_. (See Exercise ILT.M60, Exercise SLT.M60, Exercise MR.M60.)