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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].\end{equation*}

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].\end{equation*}

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

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

\begin{equation*}\begin{pmatrix} 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}\end{equation*}

\begin{equation*}\begin{pmatrix} 1 / \sqrt{5} & 2 / \sqrt{5} \\ - 2 /\sqrt{5} & 1/ \sqrt{5} \end{pmatrix}\end{equation*}

\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*}

\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*}

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

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

\(\langle {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf y}, {\mathbf x} \rangle\).

\(\langle {\mathbf x}, {\mathbf y} + {\mathbf w} \rangle = \langle {\mathbf x}, {\mathbf y} \rangle + \langle {\mathbf x}, {\mathbf w} \rangle\).

\(\langle \alpha {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf x}, \alpha {\mathbf y} \rangle = \alpha \langle {\mathbf x}, {\mathbf y} \rangle\).

\(\langle {\mathbf x}, {\mathbf x} \rangle \geq 0\) with equality exactly when \({\mathbf x} = 0\).

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

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.

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

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

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

Prove that \(SO(n)\) is a normal subgroup of \(O(n)\).

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

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

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

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

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

\(H \cap N = \{ \identity \}\);

\(HN=G\).

Show that each of the following is true.

\(S_3\) is the semidirect product of \(A_3\) by \(H = \{(1), (12) \}\).

The quaternion group, \(Q_8\), cannot be written as a semidirect product.

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

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

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

Find the rotation group of a dodecahedron.

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