Chapter 6.1, Problem 46E

For the linear transformation from Exercise $45$, let $\theta =45°$ and find the preimage of $v=\left(1,1\right)$.

45. Let $T$ be a linear transformation from $R^2$ into $R^2$ such that $T\left(x,y\right)=\left(x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta \right)$. Find (a) $T\left(4,4\right)$ for $\theta =45°$,

(b) $T\left(4,4\right)$ for $\theta =30°$, and (c) $T\left(5,0\right)$ for $\theta =120°$.