Hey can some help me with these kinds of questions have a lot of them and I can't figure out how to do them thanks
Let $v = (cos \theta, sin\theta)^{T} \in R^2$ for some angle $\theta \in 2^{R}$, and let $P_{v}$ denote the orthogonal projection corresponding to v. Find the matrix representation of $P_{v}$ with respect to the standard basis for $R^{2}$.
Hint: The orthogonal projection onto $v$ is defined by $w \mapsto \dfrac{\left<v,w\right>}{|v|^2}v = \dfrac{1}{|v|^2}v\left(v^T w\right) = \left(\dfrac{1}{|v|^2}vv^T\right) w$