Linear Algebra Done Right · Problem 3C.3

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrices-linear-maps

Problem

Suppose

v_1, \ldots, v_n

is a basis of

V

and

w_1, \ldots, w_m

is a basis of

W

. (a) Show that if

S, T \in \mathcal{L}(V, W)

, then

\mathcal{M}(S + T) = \mathcal{M}(S) + \mathcal{M}(T)

. (b) Show that if

\lambda \in \mathbf{F}

and

T \in \mathcal{L}(V, W)

, then

\mathcal{M}(\lambda T) = \lambda \mathcal{M}(T)

. This exercise asks you to verify 3.35 and 3.38.