<s>[INST] Find the linear transformation of a matrix knowing 4 linear transformations I'm stuck with an exercise where they give me 4 linear transformations $T:M_{2\times2}\to\Bbb R$ for 4 matrices 2x2 and then they ask me for the linear transformation of a fifth matrix.
$$ T
  \begin{pmatrix}
    1 &0\\
    0 & 0\\
  \end{pmatrix}
=3,
T\begin{pmatrix}
    0 &1\\
    1 & 0\\
  \end{pmatrix}
=-1,
T
  \begin{pmatrix}
    1 &0\\
   1 & 0\\
  \end{pmatrix}
=0,
T
  \begin{pmatrix}
    0 &0\\
    0 & 1\\
  \end{pmatrix}
=0,
T
  \begin{pmatrix}
    a &b\\
    c & d\\
  \end{pmatrix}
=?$$
and I have worked with this situation for $T:\Bbb R^n \to\Bbb R^m$ but never with matrices so I have the doubt that how is supposed to find 4 constants that will multiply de 4 matrices I know to compose the fifth matrix to finally can find the linear transformation of this last one. (I have tried adding up the 4 matrices as follows)
$$ 
  \begin{pmatrix}
    1 &0\\
    0 & 0\\
  \end{pmatrix}
+
\begin{pmatrix}
    0 &1\\
    1 & 0\\
  \end{pmatrix}
+
  \begin{pmatrix}
    1 &0\\
   1 & 0\\
  \end{pmatrix}
+
  \begin{pmatrix}
    0 &0\\
    0 & 1\\
  \end{pmatrix}=
  \begin{pmatrix}
    a &b\\
    c & d\\
  \end{pmatrix}
$$
\begin{cases}
2=a  &  \\
1=b & \\
2=c\\
1=d\\
\end{cases}
and then use the coefficients
$$ 
  2\begin{pmatrix}
    1 &0\\
    0 & 0\\
  \end{pmatrix}
+
1\begin{pmatrix}
    0 &1\\
    1 & 0\\
  \end{pmatrix}
+
  2\begin{pmatrix}
    1 &0\\
   1 & 0\\
  \end{pmatrix}
+
  1\begin{pmatrix}
    0 &0\\
    0 & 1\\
  \end{pmatrix}=
  \begin{pmatrix}
    a &b\\
    c & d\\
  \end{pmatrix}
$$
and I just want to know if I continue in this way or I have committed any mistakes, thanks for your time <3
 [/INST] What is given can be translated as follows. Denote, in a standard way,
$$E_1=\begin{pmatrix}1&0\\0&0 \end{pmatrix},\quad E_2=\begin{pmatrix}0&1\\0&0 \end{pmatrix}, \quad E_3=\begin{pmatrix}0&0\\1&0 \end{pmatrix},\quad E_4=\begin{pmatrix}0&0\\0&1\end{pmatrix}.$$
We have $\:M=\begin{pmatrix}a&b\\c&d \end{pmatrix}=aE_1+bE_2+cE_3+dE_4$, so
$$T(M)=aT(E_1)+bT(E_2)+cT(E_3)+dT(E_4).$$
Now, we are given that $T(E_1)=3,\enspace T(E_2)+T(E_3)=-1,\enspace T(E_1)+T(E_3)=0, \enspace T(E_4)=0$. Can you proceed?
</s>