I have a problem finding the matrix of a linear transformation for:
$T(x_1 + x_2)= x_1, T(x_1 - x_2) = x_2$
I tried to solve this by representing the first transformation as:
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
and multiplying by the coefficients of the first transformation, s.t the resulting matrix is
\begin{pmatrix} a + b \\ c + d \end{pmatrix}
which is equal to:
\begin{pmatrix} 1 \\ 0 \end{pmatrix}
repeating the same process for the second transformation, I obtained the following system of equations:
\begin{array}{lcl} a +b = 1, a - b = 0 \\ c + d = 0, c-d=1 \end{array}
to which I get:
\begin{pmatrix} .5 + .5 \\ .5 - .5 \end{pmatrix}
Is this correct?