Let us define a linear transformation $T:\mathbb{R}^3\to\mathbb{R}^3$ by $T\begin{pmatrix}x\\y\\z\\\end{pmatrix} = \begin{pmatrix}x+y\\y\\2x+z\\\end{pmatrix}$.
We can say that the matrix of this linear transformation with respect to the standard basis $\mathscr{B} = \{e_1,e_2,e_3\}$ is $Mat_{\mathscr{B},\mathscr{B}}=\left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array} \right) $.
I have a question about the notation for a similar application, but where we have a polynomial vector space.
Now let us define another linear transformation $R:\mathcal{P}_2\to\mathcal{P}_2$ by $R(a+bx+cx^2)=(a+b+c)+(2a+3c)x+(2a+2b+c)x^2$
Say we have the basis $\mathscr{C}=\{1,x,x^2\}$, which spans the space of polynomials of degree at most 2.
How would we write $Mat_{\mathscr{C},\mathscr{C}}$? This is purely a notational question, as we do not write the basis vectors for the polynomials as column vectors like we do for $\mathbb{R}^n$.
I know that we would perform the following calculations:
$R(1)=1+2x+2x^2=1(1)+2(x)+2(x^2)$
$R(x)=1+2x^2=1(1)+0(x)+2(x^2)$
$R(x^2)=1+3x+x^2=1(1)+3(x)+1(x^2)$
So each transformation is written as a sum of the basis vectors (or basis polynomials?).
So would we write $Mat_{\mathscr{C},\mathscr{C}}=\left( \begin{array}{ccc} 1 & 2 & 2 \\ 1 & 0 & 2 \\ 1 & 3 & 1 \end{array} \right) $
I have perhaps answered my own question - could anyone verify that this is correct?