Consider the vector space $\mathbb{R}^2$ with the standard inner product given by $ \langle(a, b), (c, d)\rangle = ac + bd$. (This is just the dot product.)
(a) Let $\theta \in [0,2\pi)$ and let T : $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation whose standard matrix representation A(T) is the clockwise rotation by $\theta$ is $[\cos \theta -\sin\theta ; \sin\theta \cos\theta]$.
Show that for all vectors v and w, we have $$\langle T(v),T(w) \rangle = \langle v,w\rangle \ \ (1)$$
(Hint: Recall that $(\cos\theta)^2 + (\sin\theta)^2 = 1)$
(b) Conversely, assume that for all vectors $v$ and $w$, we have (1). Show that there exists $\theta \in [0,2\pi)$ such that the standard matrix representation of T is AT= $[\cos \theta -\sin\theta ; \sin\theta \cos\theta]$.
In this question, I don't really understand the connection between the T(v), T(w) to the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. So I don't know where can I start
Best Answer
You want to prove that a rotation in $\mathbb{R}^2$ is a linear transformation which preserves inner products, and you want to prove some kind of converse that every linear transformation of the plane which preserves inner products is of a certain kind.
Given two column vectors $u,v \in \mathbb{R}^2$, let $u^T v$ denote their inner product. So, if $u=(a,b)^T, v=(c,d)^T$, then $u^T v = (a,b)(c,d)^T=ac+bd$. In your question, $T(v)$ denotes the image of $v$ under the linear transformation $T$. If $A$ is the matrix representing the linear transformation $T$, then $T(v)$ is equal to $Av$. So, the inner product $\langle T(v), T(u) \rangle$ is equal to $\langle Av, Au \rangle = (Av)^T (Au)$ $=v^T A^T A u$. For part (a), you can calculate $A^T A$ and verify that you get the $2 \times 2$ identity matrix.
For part (b), let $A = \left[\begin{array}{cc} a & b \\ c & d \end{array} \right]$ be the matrix representing the linear transformation $T$. We need to show that if $T$ preserves lengths, then its matrix representation $A$ has the form you mention (which implies that $T$ is a rotation). Actually, I think $T$ can also be a reflection in this case, so the exercise doesn't sound correct. (In general, a linear transformation which preserves lengths is called an orthogonal transformation, and every orthogonal transformation of the plane preserves inner products and is either a reflection or a rotation.)
In any case, for (b), you can take $u=v=(1,0)$ to get that $a^2+c^2=1$ (since $T$ preserves the length $\langle u, u \rangle$ of $u$ to be 1). Assume $a=\cos \theta$ for some $\theta$ and solve for the other variables. You can take $u=v=(1,0)$ and $u=v=(1,1)$ to get the other equations.