My book asks us to find the standard matrix $A$ for the linear transformation $T$, where $T$ is the counterclockwise rotation of $45$ degrees in $R^2$. Their solution starts by saying:
$T(x,y) = (\cos(45^\circ) x - \sin(45^\circ) y, \ \sin(45^\circ) x + \cos(45^\circ) y)$
Can someone explain to me why this is?
I was able to otherwise complete the problem and find $A$ just by finding $T(1,0)$ and $T(0,1)$ using the unit circle, I don't understand this step they did. It's been a while since I had to do much with trigonometry so maybe I just need a little refresher.