# Find matrix of polynomial linear transformation relative to basis

I have been attempting quite a few of these "find the matrix from the linear transformation" problems and I have been stuck on this problem for some time. For the standard bases what I have is

\begin{bmatrix} p(1) & 0 & 0 \\ 0 & p\prime(1) & 0 \\ \\ 0 & 0 & p\prime\prime(-1) \end{bmatrix}

and here's what I have for the other basis

\begin{bmatrix} p(1) & p(1) & p(1) \\ 0 & p\prime(1) & p\prime(1) \\ \\ 0 & 0 & p\prime\prime(-1) \end{bmatrix}

Thanks in advance!

Let $T:P_3(\mathbb{R})\rightarrow\mathbb{R^3}$ be the linear transformation given by $$T(p)=(p(1), p\prime(1), p\prime\prime(-1))$$.

Find the matrix of $T$ relative to the standard basis of $P_3(\mathbb{R})$ and the basis $$v_1=(1,0,0), v_2=(1,1,0), v_3=(1,1,1)$$ of $\mathbb{R^3}$

• Why do you tell us what the standard basis of $P_3(R)$ actually is, and for each vector $p$ in that basis, tell us what you computed as $T(p)$?That wy we'll know a bit more of what you actually know how to do. (You can edit your question by clicking on "edit" just below the question, to show your partial work.) – John Hughes Oct 27 '17 at 23:42
• What are you getting stuck on? – copper.hat Oct 27 '17 at 23:50
• What is $P_3(\mathbb{R})$? Is it the vector space (algebra actually) of the polynomials of degree less than $3$, at most $3$, or something else? Telling us the "standard basis" would have clarified the question – Tancredi Oct 27 '17 at 23:54
• Agree with john hughes on this that it would be more helpful if you show what you've done. Best – vkan Oct 28 '17 at 0:16
• I have edited the post. – john fowles Oct 28 '17 at 0:42

## 1 Answer

Hint: would suggest by writing $p=ax^2+bx+c$ and seeing what happens to it by applying the transformation T.

(If the non-standard basis of $\mathbb{R}^3$ is confusing then perhaps do the problem with respect to the standard basis first).