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Exercises 12.4 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 \(SO(n)\text{?}\)
\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}
4/5 & 0 & 3 /5 \\
-3 /5 & 0 & 4 /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*}
Hint .
(a) is in
\(SO(2)\text{;}\) (c) is not in
\(O(3)\text{.}\)
4. 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.
\(\langle {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf y}, {\mathbf x} \rangle\text{.}\)
\(\langle {\mathbf x}, {\mathbf y} + {\mathbf w} \rangle = \langle {\mathbf x}, {\mathbf y} \rangle + \langle {\mathbf x}, {\mathbf w} \rangle\text{.}\)
\(\langle \alpha {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf x}, \alpha {\mathbf y} \rangle = \alpha \langle {\mathbf x}, {\mathbf y} \rangle\text{.}\)
\(\langle {\mathbf x}, {\mathbf x} \rangle \geq 0\) with equality exactly when
\({\mathbf x} = 0\text{.}\)
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
\(O(n)\text{.}\)
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
\(H \cap N = \{ \identity \}\text{;}\)
Show that each of the following is true.
\(S_3\) is the semidirect product of
\(A_3\) by
\(H = \{(1), (1 \,2) \}\text{.}\)
The quaternion group,
\(Q_8\text{,}\) cannot be written as a semidirect product.
\(E(2)\) 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 12.2.5 .
17. Determine which of the
\(17\) wallpaper groups preserves the symmetry of the pattern in
Figure 12.4.1 .
Figure 12.4.1. Lattice for Exercise 12.4.17
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.
