16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.1.11

Linear Transformations

Mathematics — Linear Algebra · zero-transformation-zero-matrix

Problem

Let

V

be the space of

n \times 1

matrices over

F

and let

W

be the space of

m \times 1

matrices over

F

. Let

A

be a fixed

m \times n

matrix over

F

and let

T

be the linear transformation from

V

into

W

defined by

T(X) = AX

. Prove that

T

is the zero transformation if and only if

A

is the zero matrix.