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The linear tranformation $L$ defined by

$L(p(x)) = 4 p^{\prime} - 4 p^{\prime\prime}$

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.