Show that T is a rotation through an angle, and find the angle $T\left[\begin{array}{c}x \\ y \end{array}\right] = \frac{1}{\sqrt{2}}\left[\begin{array}{c}x+y \\ -x+y \end{array}\right]$
This is my attempt, but I'm stuck. Can someone continue?
$T\left[\begin{array}{c} x \\ y \end{array}\right] = \left[\begin{array}{cc} \cos θ & -\sin θ \\ \sin θ & \cos θ \end{array}\right] \left[\begin{array}{c}x \\ y \end{array}\right]$
$= \left[\begin{array}{c} x\cos θ - y\sin θ \\ x\sin θ + y\cos θ \end{array}\right]$
 A: You want to find a rotation matrix that has the same effect as the matrix you have. You can then easily find the angle of rotation. We can write it like this to make it clearer (I've just re-written your first equation as a 
matrix product rather than the way you have it):
$$
\left( \begin{array}{ccc}
\cos\theta & -\sin \theta  \\
\sin\theta & \cos \theta   \end{array} \right) \left(
\begin{array}{c}
x\\
y\\
\end{array}
\right)= \left( \begin{array}{ccc}
\frac{1}{\sqrt{2}} &\frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \right) \left(
\begin{array}{c}
x\\
y\\
\end{array}
\right)
$$
so you can see we need to find the angle $\theta$ such that:
$$ \cos\theta=\frac{1}{\sqrt{2}} \;\;\text{  and  } \;\; -\sin\theta=\frac{1}{\sqrt{2}}\\
\sin\theta=-\frac{1}{\sqrt{2}} \;\;\text{  and  } \;\; \cos\theta=\frac{1}{\sqrt{2}}
$$
which is a consistent set of equations with a solution: $$\theta=\frac{\pi}{4}$$
So we can then interpret it as a rotation by an angle of $\pi/4$ radians or $45^\circ$.

Hopefully this makes it a little clearer how I got the right hand side of the first equation:
$$ T \left(\begin{array}{c}
x\\
y\\
\end{array}
\right) =\frac{1}{\sqrt{2}}\left(\begin{array}{c}
x+y\\
-x+y\\
\end{array}
\right)=\frac{1}{\sqrt{2}}\left( \begin{array}{ccc}
1 & 1  \\
-1 & 1   \end{array} \right)\left(\begin{array}{c}
x\\
y\\
\end{array}
\right)=\left( \begin{array}{ccc}
\frac{1}{\sqrt{2}} &\frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \right) \left(
\begin{array}{c}
x\\
y\\
\end{array}
\right)$$
