WeBWorK Standalone Renderer

Let $P_n$ be the vector space of polynomials $p(x)$ of degree $\leq n$ , with real coefficients.

The linear tranformation $L$ defined by $L(p(x)) = {4p'-4p''}$ maps $P_3$ into $P_2$ .

(a) Find the matrix representation $S$ of $L$ with respect to the ordered bases $E = \{x^3, x^2, x, 1 \}\quad \text{and}\quad F = \{ x^2 + x + 1, x+1, 1 \}$

$S =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$ $\left]\Rule{0pt}{3.6em}{0pt}\right.$

(b) Use Part (a) to find the coordinate vectors $[L(p(x))]_F$ of $L(p(x))$ , and $[L(g(x))]_F$ of $L(g(x))$ , with respect to the basis $F$ , where $p(x) = 15 x^3 - 5 x$ and $g(x) = x^2 - 3$ .

$[L(p(x))]_F =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$ $\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\qquad [L(g(x))]_F =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$ $\left]\Rule{0pt}{3.6em}{0pt}\right.$

You can earn partial credit on this problem.