# Tagged: cosine

## Problem 684

Let $\mathbb{R}^2$ be the vector space of size-2 column vectors. This vector space has an inner product defined by $\langle \mathbf{v} , \mathbf{w} \rangle = \mathbf{v}^\trans \mathbf{w}$. A linear transformation $T : \R^2 \rightarrow \R^2$ is called an orthogonal transformation if for all $\mathbf{v} , \mathbf{w} \in \R^2$,
$\langle T(\mathbf{v}) , T(\mathbf{w}) \rangle = \langle \mathbf{v} , \mathbf{w} \rangle.$

For a fixed angle $\theta \in [0, 2 \pi )$ , define the matrix
$[T] = \begin{bmatrix} \cos (\theta) & – \sin ( \theta ) \\ \sin ( \theta ) & \cos ( \theta ) \end{bmatrix}$ and the linear transformation $T : \R^2 \rightarrow \R^2$ by
$T( \mathbf{v} ) = [T] \mathbf{v}.$

Prove that $T$ is an orthogonal transformation.

## Problem 551

Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$.
Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$.
Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$.

Prove that
$\cos ^2\theta_1+\cdots+\cos^2 \theta_n=1.$