Consider the linear trasnformation $L:\mathbb{P^3}\rightarrow \mathbb{P^2}$ given by the formula $L(p(t))=p'(t)-tp''(t)$ Find the matrix of the linear transformation $L$ with respect to the bases $S=\{1+t^3,t+t^2,t^2−t^3,t^3\}$ and $T=\{1,t,t^2\}$. Here's what I did to attempt the problem. I found the transformation of the standard bases to be :





So I know that the matrix $A$ representing the linear transformation is :


I don't know what to do after this step. How do I find the matrix with respect to both bases $S$ and $T$? I apologize if I did something horrible wrong or if the answer is obvious!

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    $\begingroup$ Did you mean $p''(t)$ when you wrote $t''(p)$? $\endgroup$ Oct 15 '19 at 17:12
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    $\begingroup$ and $L(e_1)$ when you wrote $T(e_1)$? $\endgroup$ Oct 15 '19 at 17:13
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    $\begingroup$ That's the standard basis for $\Bbb R^4$; what's $\Bbb P^3$? $\endgroup$ Oct 15 '19 at 17:25
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    $\begingroup$ Oh, I get it now, $T$ is the basis for the codomain, so find what $L$ does to each element of $S$ and express the results as linear combinations of elements of $T$ $\endgroup$ Oct 15 '19 at 17:28
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    $\begingroup$ Do you know what $L(s)$ is for each $s\in S$? $\endgroup$ Oct 15 '19 at 17:36

$S=\{1+t^3,t+t^2,t^2−t^3,t^3\}$ is a basis of $\Bbb P^3$, the domain of $L$,

and $T=\{1,t,t^2\}$ is a basis of $\Bbb P^2$, the codomain of $L$.

$L$ maps the elements of $S$ to $-3t^2, 1, 3t^2, -3t^2$, respectively,

so $A=\begin{bmatrix}0&1&0&0\\0&0&0&0\\-3&0&3&-3\end{bmatrix}.$

Note that $A$ has the proper size for mapping a $4$-dimensional space to a $3$-dimensional space.

  • $\begingroup$ I was going about solving the question in a completely wrong way! Thank you! $\endgroup$ Oct 16 '19 at 10:25

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