Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 49

Exercises

Exercise

Suppose that the pieces of a jigsaw puzzle have roughly a rectangular form, where one side is visibly longer than the other, and on every side there is either a tab or a blank. How many types of pieces exist?

Exercise

Determine the order of the plane rotation about ${}291$ degree.

Exercise

How many elements does the subgroup of the rotation group ${}\operatorname {SO} _{2}\!$ have that is generated by the rotation about ${}45$ degree, the rotation about ${}99$ degree, and the twelth part of a complete rotation.

Exercise

We consider the group of rotations at a circle that are given by multiples of the angle ${}\alpha =360/12=30$ degree. Which rotations are generators of this group?

Exercise

Consider an equilateral triangle with the origin as center and with ${}(1,0)$ as one vertex. Determine the (proper and improper) matrices that correspond to the symmetries of this triangle.

Exercise

Let ${}Q$ be the square in ${}\mathbb {R} ^{2}$ with the vertices ${}(1,0),(0,1),(-1,0),(0,-1)$.

a) Determine for every proper symmetry of this square the describing matrix with respect to the standard basis.

b) Determine for every improper symmetry of this square the describing matrix with respect to the standard basis.

c) Is the group of all proper and improper symmetries of this square commutative?

Exercise

Determine all matrices that correspond to the symmetries of a square with the vertices ${}(\pm 1,\pm 1)$. Do these matrices for every square (with the origin as center) look alike?

Exercise

We consider a rectangle in the plane, not a square, and with the properties that its center is the origin and the edges are parallel to the coordinate axes. Determine the matrices that describe the (proper and improper) symmetries of the rectangle. Establish an operation table for this symmetry group.

Exercise

What numbers appear as the order of a proper symmetry of the cube? Describe the action of every symmetry on the vertices, the edges, and the faces of the cube. Moreover, describe the action of every symmetry on the space diagonal axes, on the axes given by the centers of the faces, and on the axes given by the centers of the edges.

Exercise

Determine the matrices of the four symmetries of a cube with the vertices ${}(\pm 1,\pm 1,\pm 1)$ that map ${}(1,0,0)$ to ${}(0,0,-1)$.

Exercise

How many possibilities do exist to number the faces of a cube with the numbers ${}1$ to ${}6$ such that the sum of the numbers on opposite faces is ${}7$?

Exercise

Suppose that the vertices ${}(\pm 1,\pm 1,\pm 1)$ of a cube are denoted by ${}1,2,3,\ldots ,8$. Determine the value tables that describe the permutations of the vertices induced by the following (proper or improper) symmetries of the cube:

a) ${}{\begin{pmatrix}1&0&0\\0&0&1\\0&-1&0\end{pmatrix}}$,

b) ${}{\begin{pmatrix}-1&0&0\\0&0&-1\\0&-1&0\end{pmatrix}}$,

c) ${}{\begin{pmatrix}0&0&1\\0&1&0\\-1&0&0\end{pmatrix}}$.

Exercise

Let ${}W$ be the cube with the vertices ${}(\pm 1,\pm 1,\pm 1)$. We fix an axis through two midpoints of opposite edges. Which symmetries of the cube can be described as a rotation around this axis? How do the corresponding matrices look like? What happens to the vertices of the cube?

Exercise

Determine the coordinates of the vertices of a tetrahedron fulfilling the following properties: the origin is the center; the four vertices have distance one to the origin; the point ${}(0,0,1)$ is a vertex; there is another vertex with coordinates of the form ${}(u,0,v)$.

Hand-in-exercises

Exercise (2 marks)

How many elements does the subgroup of the rotation group ${}\operatorname {SO} _{2}\!$ have that is generated by the rotation about ${}51$ degree, the rotation about ${}99$ degree, and the seventh part of a complete rotation.

Exercise (2 marks)

Let ${}W$ be the cube with the vertices ${}(\pm 1,\pm 1,\pm 1)$. Let ${}\varphi$ be a rotation around ${}120$ degree around the space diagonal through ${}(1,1,1)$ and ${}(-1,-1,-1)$. Determine the plane equations for those planes that are determined by three vertices that are moved to each other by this rotation.

Exercise (5 marks)

Describe for each symmetry of the tetrahedron depicted above a suitable matrix representation.

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