0
$\begingroup$

Let L be the operator on $P_3$ defined by

$L(P(x))=xp'(x)+p''(x)$

$(a)$ Find the matrix $A$ representing $L$ with respect to $\left[ 1,x,x^2\right].$

$(b)$ Find the matrix $B$ representing $L$ with respect to $\left[ 1,x,1+x^2\right].$

$(c)$ Find the matrix $S$ such that $B=S^{-1}AS$

$(d)$ If $p(x)= a_0+a_{1}x+a_2(1+x^2),\text{calculate } L^n(p(x))n$

EDIT

$$(A) \\ L(1) = 0 \\ L(x) =x+0 \\ L(x^2)=2x^2+2 \\ \Rightarrow \begin{bmatrix} 0 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix} $$

$$(B) \\ L(1) = 0 \\ L(x) =x+0 \\ L(1+x^2)=2x^2+2 \\ \Rightarrow \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix} $$

For part (c) how does one find S? I know one must use the transition matrix from

$\left[ 1,x,x^2\right].$ to$\left[ 1,x,1+x^2\right].$

$$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix}^{-1}$$

  • 1
    The last matrix you wrote has a zero column, so it is definitly not invertable. As you have said, S is the matrix of change of basis (transition matrix) so its columns consist out of the coordinates of the second basis with respect to the first basis – 2017-01-29
  • 0
    Student What do they consist of? – 2017-01-29
  • 0
    Sorry, i'm typing on mobile, i have edited my comment – 2017-01-29

0 Answers 0