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Exercises 15.4 Exercises

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

(a) is in \(SO(2)\text{;}\) (c) is not in \(O(3)\text{.}\)

4.

Determine the symmetry group of each of the figures in Figure 15.25.

There are two figures in the top row and one figure on bottom. The top left figure, a, is a rectangle. Inside the rectangle is an oval in the lower left and a solid circle in the upper right. The top right figure, c, is there intersecting circles of the same radii. The bottom middle figure, b, is a large square with the two diagonals. The midpoints of the large square are the vertices of a smaller inscribed square.
Figure 15.25.
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 \({\mathbf y} = 0\text{.}\)

Hint

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

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.

Hint

Use the unimodular matrix

\begin{equation*} \begin{pmatrix} 5 & 2 \\ 2 & 1 \end{pmatrix}\text{.} \end{equation*}
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 \(O(n)\text{.}\)

Hint

Show that the kernel of the map \(\det : O(n) \rightarrow {\mathbb R}^*\) is \(SO(n)\text{.}\)

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 \(O(n)\text{.}\)

Hint

True.

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 15.16.

17.

Determine which of the 17 wallpaper groups preserves the symmetry of the pattern in Figure 15.26.

Hint

\(p6m\)

A lattices of hexagons. Each hexagon is divided into three rhombuses.
Figure 15.26.
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.