A First Course in Linear Algebra · Problem MR.T40

Representations — Robert Beezer

Mathematics — Linear Algebra · linear-transformations-isomorphic-matrices

Problem

Theorem VSLT defines the vector space

\mathcal{LT}(U,V)

containing all linear transformations with domain

U

and codomain

V

. Suppose

\dim(U) = n

and

\dim(V) = m

. Prove that

\mathcal{LT}(U,V)

is isomorphic to

M_{mn}

, the vector space of all

m \times n

matrices (Example VSM). (Hint: we could have suggested this exercise in Chapter LT, but have postponed it to this section. Why?)