The linear tranformation $L$ defined by

maps $P_4$ into $P_3$.

(a) Find the matrix representation of $L$ with respect to the ordered bases

$E = \lbrace x^3, x^2, x, 1 \rbrace$ and $F = \lbrace x^2 + x + 1, x+1, 1 \rbrace$

$S =$
 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

(b) Use Part (a) to find the coordinate vectors of $L(p(x))$ and $L(g(x))$ where $p(x) = 15 x^3 - 5 x$ and $g(x) = x^2 - 3$.

$[L(p(x))]_F =$
 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$
$[L(g(x))]_F =$
 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

You can earn partial credit on this problem.