<s>[INST] Linear algebra mapping question Does there exist a matrix $A$ such that 
$$
A\left( \begin{array}{ccc}
1 \\
0 \\
0 \\
0  \\
\end{array} \right) = \left( \begin{array}{ccc}
1 \\
0 \\
0 \\
0  \\
\end{array} \right)$$
$$
A\left( \begin{array}{ccc}
0 \\
1 \\
0 \\
0  \\
\end{array} \right) = \left( \begin{array}{ccc}
0 \\
1 \\
0 \\
0  \\
\end{array} \right)$$
$$
A\left( \begin{array}{ccc}
0 \\
0 \\
1 \\
0  \\
\end{array} \right) = \left( \begin{array}{ccc}
0 \\
1 \\
0 \\
0  \\
\end{array} \right)$$
$$
A\left( \begin{array}{ccc}
0 \\
0 \\
0 \\
1  \\
\end{array} \right) = \left( \begin{array}{ccc}
0 \\
0 \\
1 \\
0  \\
\end{array} \right)$$
if so, what?
 [/INST] If $A$ exists, $A$ satisfies
$$A\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\end{pmatrix}=
\begin{pmatrix}
1&0&0&0\\
0&1&1&0\\
0&0&0&1\\
0&0&0&0\end{pmatrix}$$
So
$$A=
\begin{pmatrix}
1&0&0&0\\
0&1&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix}$$
</s>