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## 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/ \sqrt{5} & 0 & 3 / \sqrt{5} \\ -3 / \sqrt{5} & 0 & 4 / \sqrt{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. 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 $H = \{(1), (12) \}\text{.}$

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. 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.