Question

3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix representation of L with respect to your chosen basis. (d) Show L : V → V is invertible ·Let O : 1. → 11. be the zero linear transfrination defined by O(r)-O" for any r e l Show that the matrix representation for O with respect to any ordered bases for V' and ll' is the m x n zero matrix, where n-dim(V) and m = dim(W) 5. Let 1 : V → V le the linear transformation defined by I(e) r for any r € 1". Show that the matrix represeutation for I with respect to any ordered basis for V is I where

Answer 1

the vertrY し(1)-- 2- eigen フも . C+らん尤し ヅラ-J LV

Then ttin L()- t next 3'

K d ho a レ. w· 0 nr n V將 UV 00, 10 5ル 00.0