Linear map : Matrix of transformation We have the matrices \begin{equation*}A_1:=\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}, \ \ A_2:=\begin{pmatrix}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{pmatrix}\end{equation*} 
We consider the linear maps $F_i:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $\vec{x}\mapsto A_i\vec{x}$ for $i=1,2$. 
The maps $F_2\circ F_1$ and $F_1\circ F_2$ aren't equal, are they? 
To define the matrix of the correspondig map do we multiply the two matrices? 
 A: For
$F_i:\mathbb{R}^2\rightarrow \mathbb{R}^2$ ,
$F_1 \circ F_2$ and $F_2 \circ F_1$ are different. 
Matrix is a shorthand(/representation) of linear transformation, so I'll suggest you to multiply both matrices in respective orders and compute where they send $(x,y)$ so that you can observe that composition of linear transformation is multiplication of respective matrices!
A: That's correct.
A point worth noting:
$$\begin{bmatrix}
\color{red}{\lambda_1} & 0 \\
0 & \color{blue}{\lambda2}
\end{bmatrix}\cdot
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}=
\begin{bmatrix}
\color{red}{\lambda_1} a & \color{red}{\lambda_1} b \\
\color{blue}{\lambda_2} c & \color{blue}{\lambda_2} d
\end{bmatrix}$$
$$
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}\cdot
\begin{bmatrix}
\color{red}{\lambda_1} & 0 \\
0 & \color{blue}{\lambda2}
\end{bmatrix}=
\begin{bmatrix}
\color{red}{\lambda_1} a & \color{blue}{\lambda_2} b \\
\color{red}{\lambda_1} c & \color{blue}{\lambda_2} d
\end{bmatrix}$$
Note how rows are multiplied by $\lambda_1, \lambda_2$ when you multiply by a diagonal matrix on the left, while columns are multiplied $\lambda_1, \lambda_2$ when you multiply by a diagonal matrix on the right.
