Exercises

01

Rotation matrix

Using geometry and trig sum identities, verify that this is the correct formula for rotation of the point $(x,y)$ counterclockwise about the origin by the angle $\theta$.

02

Computing a projection matrix

Find the matrix of the projection of ${\mathbb{R}}^2$ in the direction of the vector $\mathbf{v}=(3,-4)$. (First compute the unit vector.)

03

Double projections

Consider a unit vector $\mathbf{v}$ and another vector $\mathbf{u}$. Show that ${\mathrm{proj}}_{\mathbf{v}}({\mathrm{proj}}_{\mathbf{v}}(\mathbf{u}))={\mathrm{proj}}_{\mathbf{v}}(\mathbf{u})$. Do this by applying the matrix of the projection of $\mathbf{v}$ to the vector $\mathbf{u}$ twice in sequence (meaning you apply the matrix again on the output vector of the first application of the matrix). To simplify the result, remember that $\mathbf{v}\cdot \mathbf{v}=1$ because $\mathbf{v}$ is a unit vector.

04

Computing a reflection matrix

Compute the matrix representing 3D reflection in the plane perpendicular to the vector $(2,3,6)$. (First find the unit vector!)

05

Finding a permutation matrix

Find the permutation matrix that acts as follows:

$$\left(\begin{array}{c}{a}_2\\ a_4\\ a_1\\ a_5\\ a_3\end{array}\right)⟼\left(\begin{array}{c}{a}_1\\ a_2\\ a_3\\ a_4\\ a_5\end{array}\right).$$