Will someone help me with this problem about linear transformation. Let $\displaystyle T:R^2 \rightarrow R^2$ defined to be the transformation of rotation of the plane by angle $\displaystyle \theta$ around 0.

1) Show that T is linear.

2) Compute $\displaystyle [T]_{\beta}$ given $\displaystyle \beta=\{(1,0),(0,1)\}$

3) Consider the standard inner product <u,v> on $\displaystyle R^2$. Show that it is invariant under the transformation T.

For part 1) I tried to formulate T after drawing the picture I have something like $\displaystyle T(a_1,a_2)=(rcos(\alpha+\theta),rsin(\alpha+\theta ))$, here $\displaystyle r=\sqrt{{a_1}^2+{a_2}^2}$ then I rewrite this in term of $\displaystyle \theta, a_1,a_2$, but my professor told me this is not how he wants it. He wants me to draw some paralellogram and do it like a geometric proof, which I have no idea. I tried to formulate this because I want to show T(cx+y)=cT(x)+T(y).

For part 2) he also wants me to do it geometrically. I really don't know how to di it. Can someone help please?