> 
> Let $T:\mathbb{R}^2\to \mathbb{R}^2$ be a linear transformation such that
>  $$(a,b)\longmapsto (a+b, a-b)$$
>  Find all the eigenvalues and, for each eigenvalue, find the corresponding eigenspace.
> 
> 
> 

My attempt:

I don't know if there is a normal procedure to find the matrix of a linear transformation, but I just "back filled" the entry values to make it work. So I have
$$
\begin{pmatrix}
 1 & 1 \\
 1 & -1 \\
\end{pmatrix}
\begin{pmatrix}
 a \\
 b \\
\end{pmatrix}=
\begin{pmatrix}
 a+b \\
 a-b \\
\end{pmatrix}
$$
So, denoting the matrix as $A$, I used the characteristic polynomial
$$ det(A-\lambda I)=
\begin{pmatrix}
1-\lambda & 1 \\
1 & -1-\lambda \\
\end{pmatrix}=0
$$
$\implies -(1-\lambda)^2-1=0\implies \lambda= 1+i$ or $1-i$.

pluging the former value into the matrix I solve for
$$
\begin{pmatrix}
-i & 1\\
1 & -2-i \\
\end{pmatrix}
\begin{pmatrix}
a \\
b \\
\end{pmatrix}=
\begin{pmatrix}
0 \\
0 \\
\end{pmatrix}
$$
Which generates the system of equations
$$-ia+b=0, \quad a-2b-ib=0 $$
But solving the system gives me $a=b=0$. 

There is a previous problem where I got the same thing, so I'm wondering if I am doing something wrong. Any help is much appreciated.