# Matrix of a linear transformation (notation)

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?

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If you have a ordered base $\{v_1,\ldots,v_n\}$ for a vector space, in general we denote the vector $w=a_1v_1+\cdots+a_nv_n$ only by its coefficients, that is, $w=(a_1,\ldots,a_n)$. So, the construction of the matrix of a linear map is the same.