Need help figuring out the transformation matrix in respect to the standard basis

Question

So I'm giving this function: $V = Pol_2(\mathbb{R})$

s: V $\rightarrow$ V, $\space$ s(1) := 1 $\space$ $\space$ s(x) := x - 1 $\space$ $\space$ s($x^2$) := $(x - 1)^2$

I have to give the transformation matrix in respect to the basis {1, x, $x^2$}.

As far as I know, the transformation matrix can be written by plugging in each Vector in the matrix, and using the result as a column vector in the matrix.

So as follows:

s(1) = 1 = (1, 0, 0)

s(x) = x - 1 = (-1, 1, 0)

s($x^2$) = $x^2$ - 2x + 1 = (1, -2, 1)

So then my matrix should be like this:

$\begin{bmatrix} 1&-1&1\\ 0&1&-2\\ 0&0&1\\ \end{bmatrix}$

But this does not add up when I try multiplying this matrix with my standard basis vector as follows:

$\begin{bmatrix} 1&-1&1\\ 0&1&-2\\ 0&0&1\\ \end{bmatrix}$ * $\begin{bmatrix}1\\ x\\ x^2\\ \end{bmatrix}$ = $\begin{bmatrix}1 -x + x^2\\ x - 2x^2\\ x^2\\ \end{bmatrix}$

What am I doing wrong here?

Thanks!

Answer 1

When performing the matrix multiplication in order to see the result of the linear transformation, you must use the vector with respect to the basis. You deduced correctly that the matrix for $s(x)$ is: \begin{bmatrix} 1 & -1 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{bmatrix}

You now have to use the correct vector for multiplication. If you want to know $s(1 + x + x^2)$, you first have to find the vector of $1 + x + x^2$ with respect to the ordered basis $(1, x, x^2)$. This vector is: \begin{bmatrix} 1 \\ 1\\ 1 \end{bmatrix} If we now perform the multiplication we get:

$\begin{bmatrix} 1 & -1 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{bmatrix} * \begin{bmatrix} 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix} 1 & -1 & 1\end{bmatrix}$

This corresponds to the value $1 -x + x^2$. If we evaluate $s(1 + x + x^2)$ we see that: $s(1 + x + x^2) = s(1) + s(x) + s(x^2) = 1 + x - 1 + x^2 - 2x + 1 = x^2 - x + 1$

So both methods give us the same answer. So to answer your question: you shouldn't directly input the polynomial, but you should use the vector with respect to the basis.