Computing the matrix of the linear map $X\rightsquigarrow AXA^t$, where $A=\left[\begin{smallmatrix}2&1\\0&1\end{smallmatrix}\right]$ 
Let $V$ be the vector space of real $2\times 2$ symmetric matrices
  $X=\begin{bmatrix}x&y\\y&z\end{bmatrix}$ and let
  $A=\begin{bmatrix}2&1\\0&1\end{bmatrix}$. Determine the matrix of the
  linear operator on $V$ defined by $X\rightsquigarrow AXA^t$ with
  respect to a suitable basis.

So far, I got $AXA^t=\begin{bmatrix}4x+4y+z&2y+z\\2y+z&z\end{bmatrix}$
I need to find a matrix $C$ such that $CX=AXA^t$, but not find $C$. Am I doing anything wrong. Please someone tell me what is wrong here.
 A: First, some generalities.  The structure of a vector space is totally determined by its dimension.  Given a real vector space $V$ of dimension $n$, then $V$ is isomorphic to $\mathbb{R}^n$, where the isomorphism is given by any choice of basis for $V$.  More explicitly, given a basis $\newcommand{\B}{\mathcal{B}}  \newcommand{\R}{\mathbb{R}} \B = \{v_1, \ldots, v_n\}$ for $V$, we get an isomorphism given by the coordinate map
\begin{align*}
\varphi_\B: V &\to \R^n\\
x &\mapsto [x]_\B
\end{align*}
where $[x]_\B \in \R^n$ is the coordinate vector of the vector $x \in V$, whose entries are the unique weights $c_1, \ldots, c_n \in \R$ expressing $x$ as a linear combination of $v_1, \ldots, v_n$.  That is,
$$
[x]_\B =
\begin{pmatrix}
c_1\\ \vdots\\ c_n
\end{pmatrix}
$$
where
$$
x = c_1 v_1 + \cdots + c_n v_n \, .
$$
Once we've chosen a basis and identified with $\R^n$, a linear map $T: V \to V$ can be expressed as left-multiplication by some matrix $[T]_\B$.  This is captured by the following commutative diagram.
$\hspace{6.5cm}$
As for computing $[T]_\B$, one can show that
$$
[T]_\B =
\begin{pmatrix}
| & & |\\
[T(v_1)]_\B & \cdots & [T(v_n)]_\B\\
| & & |
\end{pmatrix}
$$
where as before $\B = \{v_1, \ldots, v_n\}$.
Returning to your problem, you have found a basis for $V$, and also determined the general formula for $T$.  Now you just have to apply $T$ to the basis vectors, write each image as a linear combination of the basis vectors, and use these as the columns of $[T]_\B$.  For instance, your first basis vector is $v_1 = \begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}$ and
$$
T(v_1) = \begin{pmatrix}4 & 0\\ 0 & 0\end{pmatrix} = 4 v_1
$$
so $[T(v_1)]_\B = \begin{pmatrix}4\\ 0\\ 0\end{pmatrix}$.  Can you compute the remaining columns?
