Kunze Linear Algebra · Problem 3.2.12

Linear Transformations

Mathematics — Methods · invertibility-left-multiplication

Problem

Let

p

m

, and

n

be positive integers and

F

a field. Let

V

be the space of

m \times n

matrices over

F

and

W

the space of

p \times n

matrices over

F

. Let

B

be a fixed

p \times m

matrix and let

T

be the linear transformation from

V

into

W

defined by

T(A) = BA

. Prove that

T

is invertible if and only if

p = m

and

B

is an invertible

m \times m

matrix.