## Exercises12.4Exercises

###### 1.

Prove the identity

\begin{equation*} \langle {\mathbf x}, {\mathbf y} \rangle = \frac{1}{2} \left[ \|{\mathbf x} + {\mathbf y}\|^2 - \|{\mathbf x}\|^2 - \| {\mathbf y}\|^2 \right]\text{.} \end{equation*}
###### 2.

Show that $$O(n)$$ is a group.

###### 3.

Prove that the following matrices are orthogonal. Are any of these matrices in $$SO(n)\text{?}$$

1. \begin{equation*} \begin{pmatrix} 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \end{equation*}
2. \begin{equation*} \begin{pmatrix} 1 / \sqrt{5} & 2 / \sqrt{5} \\ - 2 /\sqrt{5} & 1/ \sqrt{5} \end{pmatrix} \end{equation*}
3. \begin{equation*} \begin{pmatrix} 4/5 & 0 & 3 /5 \\ -3 /5 & 0 & 4 /5 \\ 0 & -1 & 0 \end{pmatrix} \end{equation*}
4. \begin{equation*} \begin{pmatrix} 1/3 & 2/3 & - 2/3 \\ - 2/3 & 2/3 & 1/3 \\ 2/3 & 1/3 & 2/3 \end{pmatrix} \end{equation*}
###### 4.

Determine the symmetry group of each of the figures in Figure 12.25.

###### 5.

Let $${\mathbf x}\text{,}$$ $${\mathbf y}\text{,}$$ and $${\mathbf w}$$ be vectors in $${\mathbb R}^n$$ and $$\alpha \in {\mathbb R}\text{.}$$ Prove each of the following properties of inner products.

1. $$\langle {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf y}, {\mathbf x} \rangle\text{.}$$

2. $$\langle {\mathbf x}, {\mathbf y} + {\mathbf w} \rangle = \langle {\mathbf x}, {\mathbf y} \rangle + \langle {\mathbf x}, {\mathbf w} \rangle\text{.}$$

3. $$\langle \alpha {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf x}, \alpha {\mathbf y} \rangle = \alpha \langle {\mathbf x}, {\mathbf y} \rangle\text{.}$$

4. $$\langle {\mathbf x}, {\mathbf x} \rangle \geq 0$$ with equality exactly when $${\mathbf x} = 0\text{.}$$

5. If $$\langle {\mathbf x}, {\mathbf y} \rangle = 0$$ for all $${\mathbf x}$$ in $${\mathbb R}^n\text{,}$$ then $${\mathbf y} = 0\text{.}$$

###### 6.

Verify that

\begin{equation*} E(n) = \{(A, {\mathbf x}) : A \in O(n) \text{ and } {\mathbf x} \in {\mathbb R}^n \} \end{equation*}

is a group.

###### 7.

Prove that $$\{ (2,1), (1,1) \}$$ and $$\{ ( 12, 5), ( 7, 3) \}$$ are bases for the same lattice.

###### 8.

Let $$G$$ be a subgroup of $$E(2)$$ and suppose that $$T$$ is the translation subgroup of $$G\text{.}$$ Prove that the point group of $$G$$ is isomorphic to $$G/T\text{.}$$

###### 9.

Let $$A \in SL_2({\mathbb R})$$ and suppose that the vectors $${\mathbf x}$$ and $${\mathbf y}$$ form two sides of a parallelogram in $${\mathbb R}^2\text{.}$$ Prove that the area of this parallelogram is the same as the area of the parallelogram with sides $$A{\mathbf x}$$ and $$A{\mathbf y}\text{.}$$

###### 10.

Prove that $$SO(n)$$ is a normal subgroup of $$O(n)\text{.}$$

###### 11.

Show that any isometry $$f$$ in $${\mathbb R}^n$$ is a one-to-one map.

###### 12.

Prove or disprove: an element in $$E(2)$$ of the form $$(A, {\mathbf x})\text{,}$$ where $${\mathbf x} \neq 0\text{,}$$ has infinite order.

###### 13.

Prove or disprove: There exists an infinite abelian subgroup of $$O(n)\text{.}$$

###### 14.

Let $${\mathbf x} = (x_1, x_2)$$ be a point on the unit circle in $${\mathbb R}^2\text{;}$$ that is, $$x_1^2 + x_2^2 = 1\text{.}$$ If $$A \in O(2)\text{,}$$ show that $$A {\mathbf x}$$ is also a point on the unit circle.

###### 15.

Let $$G$$ be a group with a subgroup $$H$$ (not necessarily normal) and a normal subgroup $$N\text{.}$$ Then $$G$$ is a semidirect product of $$N$$ by $$H$$ if

• $$H \cap N = \{ \identity \}\text{;}$$

• $$HN=G\text{.}$$

Show that each of the following is true.

1. $$S_3$$ is the semidirect product of $$A_3$$ by $$H = \{(1), (1 \,2) \}\text{.}$$

2. The quaternion group, $$Q_8\text{,}$$ cannot be written as a semidirect product.

3. $$E(2)$$ is the semidirect product of $$O(2)$$ by $$H\text{,}$$ where $$H$$ consists of all translations in $${\mathbb R}^2\text{.}$$

###### 16.

Determine which of the 17 wallpaper groups preserves the symmetry of the pattern in Figure 12.16.

###### 17.

Determine which of the 17 wallpaper groups preserves the symmetry of the pattern in Figure 12.26.

###### 18.

Find the rotation group of a dodecahedron.

###### 19.

For each of the 17 wallpaper groups, draw a wallpaper pattern having that group as a symmetry group.