# Find transformation matrix $[T]_{B,B}$ representing $T$ in the basis $B$

Info provided:

$$T:P_2→P_2$$ given by $$T(p(x)) = p(kx)$$ where $$k>0$$ Find matrix $$[T]_{B,B}$$ representing $$T$$ in the basis $$B$$

Attempted Solution:

Using the standard basis for $$P_2$$, {$$1,x,x^2$$}, would $$[T]_{B,B}$$ simply be represented as $$([T(1)]_B,[T(x)]_B,[T(x^2)]_B)$$ ?

Is it that in the general case $$[T]_{W,V}$$ is from $$V$$ to $$W$$, therefore as in this case $$[T]_{B,B}$$ is simply from $$B$$ to $$B$$?

So that $$[T]_{B,B}$$ = $$\left( \begin{matrix} 1 & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k^2 \\ \end{matrix} \right)$$

Here is the detail of the case when $$B = \{1,x,x^2\}$$. You can imitate it to solve for a more general $$B$$.
We have $$\begin{equation*} \begin{cases} T(1) = 1+0x+0x^2\\ T(x) = 0+kx+0x^2\\ T(x^2) = 0+0x+k^2x^2\\ \end{cases} \end{equation*}$$ and so the matrix $$[T]_{B,B}$$ with respect to $$B = \{1,x,x^2\}$$ is of your form.
The general case is not $$[T]_{V,W}$$ but $$[T]_{B_1,B_2}$$, where $$B_1 = \{p_1,p_2,p_3\}$$ and $$B_2 = \{q_1,q_2,q_3\}$$ are bases of $$P_2$$. Just calculate the coefficients in $$\begin{equation*} \begin{cases} T(p_1) = a_{11}q_1+a_{21}q_2+a_{31}q_3\\ T(p_2) = a_{12}q_1+a_{22}q_2+a_{32}q_3\\ T(p_3) = a_{13}q_1+a_{23}q_2+a_{33}q_3 \end{cases} \end{equation*}$$ and the matrix $$[T]_{B_1,B_2}$$ is $$\begin{equation*} \begin{pmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{pmatrix}. \end{equation*}$$