10.4E: Isometries Exercises

Exercises

Throughout these exercises, \(V\) denotes a finite dimensional inner product space.

Example \(\PageIndex{1}\)

Exercise 10.4.1 Show that the following linear operators are isometries.

1. \(T: \mathbb{C} \rightarrow \mathbb{C} ; T(z)=\bar{z} ;\langle z, w\rangle=\operatorname{re}(z \bar{w})\)
2. \(T: \mathbb{R}^n \rightarrow \mathbb{R}^n ; \quad T\left(a_1, a_2, \ldots, a_n\right)=\) \(\left(a_n, a_{n-1}, \ldots, a_2, a_1\right)\); dot product
3. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} ; T\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]=\left[\begin{array}{cc}c & d \\ b & a\end{array}\right]\); \(\langle A, B\rangle=\operatorname{tr}\left(A B^T\right)\)
4. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 ; T(a, b, c)=\frac{1}{9}(2 a+2 b-c, 2 a+\) \(2 c-b, 2 b+2 c-a)\); dot product

Example \(\PageIndex{2}\)

Exercise 10.4.2 In each case, show that \(T\) is an isometry of \(\mathbb{R}^2\), determine whether it is a rotation or a reflection, and find the angle or the fixed line. Use the dot product.

1. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{r}-a \\ b\end{array}\right]\)
2. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}-a \\ -b\end{array}\right]\)
3. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{r}b \\ -a\end{array}\right]\)
4. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}-b \\ -a\end{array}\right]\)
5. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{l}a+b \\ b-a\end{array}\right]\)
6. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{l}a-b \\ a+b\end{array}\right]\)

Example \(\PageIndex{3}\)

In each case, show that \(T\) is an isometry of \(\mathbb{R}^3\), determine the type (Theorem 10.4.6), and find the axis of any rotations and the fixed plane of any reflections involved.

1. \(T\left[\begin{array}{l}a \\ b \\ c\end{array}\right]=\left[\begin{array}{r}a \\ -b \\ c\end{array}\right]\)
2. \(T\left[\begin{array}{l}a \\ b \\ c\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}\sqrt{3} c-a \\ \sqrt{3} a+c \\ 2 b\end{array}\right]\)
3. \(T\left[\begin{array}{l}a \\ b \\ c\end{array}\right]=\left[\begin{array}{l}b \\ c \\ a\end{array}\right]\)
4. \(T\left[\begin{array}{l}a \\ b \\ c\end{array}\right]=\left[\begin{array}{r}a \\ -b \\ -c\end{array}\right]\)
5. \(T\left[\begin{array}{l}a \\ b \\ c\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}a+\sqrt{3} b \\ b-\sqrt{3} a \\ 2 c\end{array}\right]\)
6. \(T\left[\begin{array}{l}a \\ b \\ c\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}a+c \\ -\sqrt{2} b \\ c-a\end{array}\right]\)

.

Example \(\PageIndex{4}\)

Let \(T : \mathbb{R}^2 \to \mathbb{R}^2\) be an isometry. A vector \(\mathbf{x}\) in \(\mathbb{R}^2\) is said to be fixed by \(T\) if \(T(\mathbf{x}) = \mathbf{x}\). Let \(E_{1}\) denote the set of all vectors in \(\mathbb{R}^2\) fixed by \(T\). Show that:

1. \(E_{1}\) is a subspace of \(\mathbb{R}^2\).
2. \(E_{1} = \mathbb{R}^2\) if and only if \(T = 1\) is the identity map.
3. \(dim \textbf{E_{1}} = 1\) if and only if \(T\) is a reflection (about the line \(E_{1}\)).
4. \(E_{1} = \{0\}\) if and only if \(T\) is a rotation (\(T \neq 1\))

Example \(\PageIndex{5}\)

Let \(T : \mathbb{R}^3 \to \mathbb{R}^3\) be an isometry, and let \(E_{1}\) be the subspace of all fixed vectors in \(\mathbb{R}^3\) (see Exercise \(\PageIndex{4}\)). Show that:

1. \(E_{1} = \mathbb{R}^3\) if and only if \(T = 1\).
2. \( dim \; \textbf{E_{1}} = 2\) if and only if \(T\) is a reflection (about the plane \(E_{1}\)).
3. \(dim \;\textbf{E_{1}} = 1\) if and only if \(T\) is a rotation (\(T \neq 1\)) (about the line \(E_{1}\)).
4. \( dim \;\textbf{E_{1}} = 0\) if and only if \(T\) is a reflection followed by a (nonidentity) rotation.

Example \(\PageIndex{6}\)

If \(T\) is an isometry, show that \(aT\) is an isometry if and only if \(a = \pm 1\)

Example \(\PageIndex{7}\)

Show that every isometry preserves the angle between any pair of nonzero vectors (see Exercise 31 Section 10.1). Must an angle preserving isomorphism be an isometry? Support your answer.

Example \(\PageIndex{8}\)

If \(T: V \rightarrow V\) is an isometry, show that \(T^2=1_V\) if and only if the only complex eigenvalues of \(T\) are 1 and -1 .

Example \(\PageIndex{9}\)

Let \(T : V \to V\) be a linear operator. Show that any two of the following conditions implies the third:

1. \(T\) is symmetric.
2. \(T\) is an involution (\(T^{2} = 1_{V}\)).
3. [Hint: In all cases, use the definition \(\langle\mathbf{v}, T(\mathbf{w}) \rangle = \langle T(\mathbf{v}), \mathbf{w} \rangle \nonumber \) of a symmetric operator. For (1) and (3) \(\Rightarrow\) (2), use the fact that, if \(\langle T^{2}(\mathbf{v}) - \mathbf{v}, \mathbf{w} \rangle = 0\) for all \(\mathbf{w}\), then \(T^{2}(\mathbf{v}) = \mathbf{v}\).]

Example \(\PageIndex{10}\)

AIf \(B\) and \(D\) are any orthonormal bases of \(V\), show that there is an isometry \(T : V \to V\) that carries \(B\) to \(D\).

Example \(\PageIndex{11}\)

Show that the following are equivalent for a linear transformation \(S : V \to V\) where \(V\) is finite dimensional and \(S \neq 0\):

1. \(\langle S(\mathbf{v}), S(\mathbf{w}) \rangle = 0\) whenever \(\langle\mathbf{v}, \mathbf{w} \rangle = 0\);
2. \(S = aT\) for some isometry \(T : V \to V\) and some \(a \neq 0\) in \(\mathbb{R}\).
3. [Hint: Given (1), show that \(\left\| S(\mathbf{e}) \right\| = \left\| S(\mathbf{f})\right\|\) for all unit vectors \(\mathbf{e}\) and \(\mathbf{f}\) in \(V\).]

Example \(\PageIndex{12}\)

Let \(S : V \to V\) be a distance preserving transformation where \(V\) is finite dimensional.

1. Show that the factorization in the proof of Theorem \(\PageIndex{1}\) is unique. That is, if \(S = S_{\mathbf{u}} \circ T\) and \(S = S_{\mathbf{u}^\prime} \circ T^\prime\) where \(\mathbf{u}\), \(\mathbf{u}^\prime \in V\) and \(T\), \(T^\prime : V \to V\) are isometries, show that \(\mathbf{u} = \mathbf{u}^\prime\) and \(T = T^\prime\).
2. If \(S = S_{\mathbf{u}} \circ T\), \(\mathbf{u} \in V\), \(T\) an isometry, show that \(\mathbf{w} \in V\) exists such that \(S = T \circ S_{\mathbf{w}}\).

Example \(\PageIndex{13}\)

Define \(T : \mathbf{P} \to \mathbf{P}\) by \(T(f) = xf(x)\) for all \(f \in \mathbf{P}\), and define an inner product on \(\mathbf{P}\) as follows: If \(f = a_{0} + a_{1}x + a_{2}x^{2} + \cdots\) and \(g = b_{0} + b_{1}x + b_{2}x^{2} + \cdots\) are in \(\mathbf{P}\), define \(\langle f, g\rangle = a_{0}b_{0} + a_{1}b_{1} + a_{2}b_{2} + \cdots\).

1. Show that \(\langle\ , \rangle\) is an inner product on \(\mathbf{P}\).
2. Show that \(T\) is an isometry of \(\mathbf{P}\).
3. Show that \(T\) is one-to-one but not onto.