Finding a matrix with respect to standard basis

[Hiche]

Homework Statement

Homework Equations

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The Attempt at a Solution

Can someone just point me how to approach this? Do we take a random second degree polynomial and input $2x + 3$ instead of $x$, then find the constants (eg. denoted by $a , b , c$) by putting the new equation equal to the standard polynomial $1 + x + x^2$?

 

[spamiam]

A good start would be to apply $T$ to the basis vectors $1, x, x^2$. What do you know about $[T]_B$? In particular, what are its columns?

 

[Hiche]

Oh, thank you. This is what I came up with so far:

$T(1) = 1$ so $(T(1))_B = \begin{pmatrix}1\\0\\0\end{pmatrix}$
$T(x) = 2x + 3$ so $(T(x))_B = \begin{pmatrix}3\\2\\0\end{pmatrix}$
$T(x^2) = 4x^2 + 12x + 9$ so $(T(x^2))_B = \begin{pmatrix}9\\12\\4\end{pmatrix}$

So, $A = [T]_B = \begin{pmatrix}1 & 3 & 9\\0 & 2 & 12\\0 & 0 & 4\end{pmatrix}$. Is this close?

 

[HallsofIvy]

Better than close- that's exactly what I get.

 

[Hiche]

Great! Now may I add a couple more questions to this?

Part b) asks to find the eigenvalues of $A$. The values I found were $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 4$.

Part c) asked to find the formula of $[T(p(x))]_B$ without justification. I am not sure how to handle this question.

 

[spamiam]

Great! Now may I add a couple more questions to this?

Part b) asks to find the eigenvalues of $A$. The values I found were $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 4$.

Part c) asked to find the formula of $[T(p(x))]_B$ without justification. I am not sure how to handle this question.

The eigenvalues look fine.

What are the B-coordinates of a polynomial $p(x) = ax^2 + bx +c$? How does the matrix $[T]_B$ transform those coordinates?