Question

Suppose that L1 : V → W and L2 : W → Z arelinear transformations and E, F, and G are orderedbases for V, W, and Z, respectively. Show that, if Arepresents L1 relative to E and F and B representsL2 relative to F and G, then the matrix C = BA representsL2 ◦ L1: V → Z relative to E and G. Hint:Show that BA[v]E = [(L2 ◦ L1)(v)]G for all v ∈ V.

Answer 1

a) v ∈ ker(L) if only if
`[V]_(E)` ∈ N(A)

b) w ∈ L(v) if and only if
`[W]_(F)` is in the column space of A

See attached

Explanation:

See attached the proof Considering the vector spaces V and W with other bases E and F respectively.

Let L be the Linear transformation form V and W and A is the matrix representing L relative to E and F