Find the matrix of $T : V\to W$ corresponding to the bases $B$ and $D$ of $V$ and $W$, respectively.

Question

$T:P_2 → P_3, T[p(x)]=xp(x); B=(1,x,x^2) $and $D=(1,x,x^2,x^3)$

I have no clue how to do this question, any help would really be appreciated! If anyone can help me do the steps, that'd be great. :)

Answer 1

$p(x) = a + bx + cx^2\\ T(p(x)) = xp(x) = ax + bx^2 + cx^3$

In matrix form:

$A\begin{bmatrix}a\\bx\\cx^2 \end{bmatrix} = \begin{bmatrix}0\\ax\\bx^2\\cx^3 \end{bmatrix}$

What is $A?$

Answer 2

Look at the images of the basis vectors in $B$ expressed in the basis $D$

$$T(1)=x=(0,1,0,0)$$

$$T(x)=x^2=(0,0,1,0)$$

$$T(x^2)=x^3=(0,0,0,1)$$

And so the matrix we are looking for is obtained by writing in columns the above row vectors

$$\begin{bmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$$