A First Course in Linear Algebra · Problem MR.T80

Representations — Robert Beezer

Mathematics — Linear Algebra · matrix-multiplication-composition-derivation

Problem

Suppose that

T: U \to V

and

S: V \to W

are linear transformations, and that

B

C

and

D

are bases for

U

V

, and

W

. Using only Definition MR define matrix representations for

T

and

S

. Using these two definitions, and Definition MR, derive a matrix representation for the composition

S \circ T

in terms of the entries of the matrices

M^T_{B,C}

and

M^S_{C,D}

. Explain how you would use this result to motivate a definition for matrix multiplication that is strikingly similar to Theorem EMP.