How form a linear mapping $f:\mathbb{R}^{4} \rightarrow \mathbb{R}^{3}$ to a matrix correctly?(solved) Given is $f: \mathbb{R}^4 \rightarrow \mathbb{R}^3$
$f(x)= \begin{pmatrix}
x_1-2x_2+x_4\\ 
-2x_1+5x_2+x_3-4x_4\\ 
x_1+2x_3-3x_4
\end{pmatrix}$
How can I form this to a matrix correctly?
We have $f: \mathbb{R}^4 \rightarrow \mathbb{R}^3$, and we have $x_1,x_2,x_3,x_4$
I think because we go to $\mathbb{R}^3$, we will only have $x_1,x_2,x_3$
So when I form a matrix, I will ignore $x_4$:
\begin{pmatrix}
 1 & -2 &  0\\ 
-2 &  5 &  1\\ 
 1 &  0 &  2
\end{pmatrix}
Is it fine like that?

Once your function is from $\Bbb R^4$ to $\Bbb R^3$ your matrix has to able to accepty a vector in $\Bbb R^4$ and give as an answer a vector in $\Bbb R^3$. It means that your matrix must have a dimension $3\times4$.
$A\begin{pmatrix}
 x_1\\ 
 x_2\\ 
 x_3\\
 x_4 
\end{pmatrix}=\begin{pmatrix}
x_1-2x_2+x_4\\ 
-2x_1+5x_2+x_3-4x_4\\ 
x_1+2x_3-3x_4
\end{pmatrix}=\begin{pmatrix}
 1 & -2 &  0 & 1\\ 
-2 &  5 &  1 &-4\\ 
 1 &  0 &  2 & -3
\end{pmatrix}.\begin{pmatrix}
 x_1\\ 
 x_2\\ 
 x_3\\
 x_4 
\end{pmatrix}$  
so
$$A=\begin{pmatrix}
 1 & -2 &  0 & 1\\ 
-2 &  5 &  1 &-4\\ 
 1 &  0 &  2 & -3\end{pmatrix}$$
