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Chapter 10: Problem 2

\(V\) denotes a finite dimensional inner product space. 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. a. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{r}-a \\\ b\end{array}\right]\) b. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}-a \\\ -b\end{array}\right]\) c. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{r}b \\\ -a\end{array}\right]\) d. \(T\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}-b \\\ -a\end{array}\right]\) e. \(T\left[\begin{array}{l}a \\\ b\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{l}a+b \\\ b-a\end{array}\right]\) f. \(T\left[\begin{array}{l}a \\\ b\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{l}a-b \\\ a+b\end{array}\right]\)

Short Answer

Expert verified
a) Reflection, fixed line y-axis; b) Rotation, 180°; c) Rotation, -90°; d) Rotation, 90°; e) Reflection, line y=x; f) Rotation, 45°.

Step by step solution

01

Verify T is an Isometry (a)

An operator \( T \) is an isometry if it preserves the dot product. Consider \( T\left[\begin{array}{l}a \ b\end{array}\right]=\left[\begin{array}{r}-a \ b\end{array}\right] \). Compute the dot product: \[ \left[\begin{array}{r}-a \ b\end{array}\right] \cdot \left[\begin{array}{l}-a \ b\end{array}\right] = (-a)^2 + b^2 = a^2 + b^2. \]The dot product \( a^2 + b^2 \) is preserved, so \( T \) is an isometry.
02

Determine Rotation or Reflection (a)

A reflection in \( \mathbb{R}^2 \) must have an eigenvalue of 1, and its transformation matrix will always have determinant -1. The matrix of the transformation is \[ \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}. \]The determinant is \(-1\), indicating that \( T \) is a reflection. This reflects over the y-axis as the fixed line.
03

Verify T is an Isometry (b)

Consider \( T\left[\begin{array}{l}a \ b\end{array}\right]=\left[\begin{array}{l}-a \ -b\end{array}\right] \). Compute the dot product: \[ \left[\begin{array}{l}-a \ -b\end{array}\right] \cdot \left[\begin{array}{l}-a \ -b\end{array}\right] = (-a)^2 + (-b)^2 = a^2 + b^2. \]Therefore, \( T \) is an isometry.
04

Determine Rotation or Reflection (b)

The matrix of the transformation is \[ \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}. \]The determinant is 1, but both eigenvalues are negative, indicating that \( T \) is a rotation by \( 180^\circ \) (a half-turn around the origin).
05

Verify T is an Isometry (c)

Consider \( T\left[\begin{array}{l}a \ b\end{array}\right]=\left[\begin{array}{r}b \ -a\end{array}\right] \). Compute the dot product: \[ \left[\begin{array}{r}b \ -a\end{array}\right] \cdot \left[\begin{array}{r}b \ -a\end{array}\right] = b^2 + (-a)^2 = a^2 + b^2. \]The dot product is preserved, so \( T \) is an isometry.
06

Determine Rotation or Reflection (c)

The matrix of the transformation is \[ \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}. \]The determinant is 1. Both eigenvalues are imaginary, indicating a rotation of \(-90^\circ\) or \(270^\circ\).
07

Verify T is an Isometry (d)

Consider \( T\left[\begin{array}{l}a \ b\end{array}\right]=\left[\begin{array}{l}-b \ -a\end{array}\right] \). Compute the dot product: \[ \left[\begin{array}{l}-b \ -a\end{array}\right] \cdot \left[\begin{array}{l}-b \ -a\end{array}\right] = (-b)^2 + (-a)^2 = a^2 + b^2. \]Thus, \( T \) is an isometry.
08

Determine Rotation or Reflection (d)

The matrix of the transformation is \[ \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix}. \]The determinant is 1. Both eigenvalues are imaginary, indicating a rotation of \( 90^\circ \).
09

Verify T is an Isometry (e)

Consider \( T\left[\begin{array}{l}a \ b\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{r}a+b \ b-a\end{array}\right] \). Compute the dot product: \[ \frac{1}{\sqrt{2}}\left[\begin{array}{r}a+b \ b-a\end{array}\right] \cdot \frac{1}{\sqrt{2}}\left[\begin{array}{r}a+b \ b-a\end{array}\right] = \frac{1}{2}((a+b)^2 + (b-a)^2) = a^2 + b^2. \]Hence, \( T \) is an isometry.
10

Determine Rotation or Reflection (e)

Finding the transformation matrix \[ \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}. \]This has a determinant of -1, so it is a reflection. The reflection is across line \( y = x \).
11

Verify T is an Isometry (f)

Consider \( T\left[\begin{array}{l}a \ b\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{r}a-b \ a+b\end{array}\right] \). Compute the dot product: \[ \frac{1}{\sqrt{2}}\left[\begin{array}{r}a-b \ a+b\end{array}\right] \cdot \frac{1}{\sqrt{2}}\left[\begin{array}{r}a-b \ a+b\end{array}\right] = \frac{1}{2}((a-b)^2 + (a+b)^2) = a^2 + b^2. \]Thus, \( T \) is an isometry.
12

Determine Rotation or Reflection (f)

Transformation matrix is \[ \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix} \], with determinant \( 1 \), indicating a rotation of angle \( 45^\circ \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isometry
In geometry, an isometry is a transformation that preserves distances between points. This concept is crucial in understanding how shapes maintain their dimensions within a space.
For a function, or operator, to qualify as an isometry, the length, or norm, of every vector must be preserved under the transformation.
Mathematically, if a transformation \( T \) satisfies \( \lVert T(v) \rVert = \lVert v \rVert \) for any vector \( v \), then \( T \) is an isometry.
In this context, the dot product plays a vital role in determining whether a transformation is an isometry.
Dot Product
The dot product is a fundamental operation in linear algebra used to measure the angle and length between vectors.
It is calculated by multiplying corresponding components of two vectors and then adding those products. For vectors \( \mathrm{a} = [a_1, a_2] \) and \( \mathrm{b} = [b_1, b_2] \), the dot product is \( a_1 \, b_1 + a_2 \, b_2 \).
It's central to the discussion of transformations, particularly when identifying isometries.
In our problems, confirming that the dot product \( a^2 + b^2 \) remains constant under transformations ensures that these operators preserve distances, classifying them as isometries.
Eigenvalues
Eigenvalues are numbers that indicate how much a transformation stretches or shrinks vectors in a given direction within a vector space.
To find eigenvalues, you solve the characteristic equation \( \det(A - \lambda I) = 0 \), where \( A \) is your transformation matrix, \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix.
In the context of rotations and reflections, eigenvalues help determine the nature of the transformation.
  • Rotation matrices typically have complex eigenvalues with magnitude one, reflecting periodic behavior.
  • Reflection matrices have real eigenvalues, one of which is often \( 1 \), indicating the axis of reflection.
This distinction helps differentiate between these two types of transformations.
Reflection
Reflection is a type of linear transformation where an image of a shape is flipped over a certain line, known as the axis of reflection.
In \( \mathbb{R}^2 \), a reflection can be represented using matrices that have a determinant of \(-1\).
These reflections reverse the orientation of the plane. For example, reflecting over the y-axis in \( \mathbb{R}^2 \), we'll flip the sign of the x component of the vector while leaving the y component unchanged.
In the exercise, matrices that resulted in a determinant of \(-1\) confirmed the presence of a reflection. For instance, the transformation across the line \( y = x \) flips vectors over that diagonal line.
Rotation
Rotation is an operation that turns every vector in space around a fixed point, typically the origin, by a specified angle.
In mathematical terms, a rotation matrix in \( \mathbb{R}^2 \) will have real entries and a determinant of \( 1 \).
These properties ensure that the transformation is distance-preserving and oriented in the same sense, such as clockwise or counter-clockwise.
A classic property of rotations is that they do not change the eigenvalues' magnitude, often resulting in eigenvalues such as \( e^{i\theta} \), where \( \theta \) is the rotation angle.
  • For instance, a rotation by \( 90^\circ \) or \( 180^\circ \) reorients vectors while maintaining their lengths, preserving dot products, confirming their status as isometries.

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Most popular questions from this chapter

Let \(U\) be a finite dimensional subspace of an inner product space \(V,\) and let \(\mathbf{v}\) be a vector in \(V\). a. Show that \(\mathbf{v}\) lies in \(U\) if and only if \(\mathbf{v}=\operatorname{proj}_{U}(\mathbf{v})\). b. If \(V=\mathbb{R}^{3}\), show that (-5,4,-3) lies in \(\operatorname{span}\\{(3,-2,5),(-1,1,1)\\}\) but that (-1,0,2) does not.

\(V\) denotes a finite dimensional inner product space. If \(B\) and \(D\) are any orthonormal bases of \(V\), show that there is an isometry \(T: V \rightarrow V\) that carries \(B\) to \(D\)

Let \(\left[x_{i}\right)=\left(x_{1}, x_{2}, \ldots\right)\) denote a sequence of real numbers \(x_{i}\), and let \(V=\left\\{\left[x_{i}\right) \mid\right.\) only finitely many \(\left.x_{i} \neq 0\right\\}\) Define componentwise addition and scalar multiplication on \(V\) as follows: \(\left[x_{i}\right)+\left[y_{i}\right)=\left[x_{i}+y_{i}\right),\) and \(a\left[x_{i}\right)=\left[a x_{i}\right)\) for \(a\) in \(\mathbb{R}\) Given \(\left[x_{i}\right)\) and \(\left[y_{i}\right)\) in \(V,\) define \(\left\langle\left[x_{i}\right),\left[y_{i}\right)\right\rangle=\sum_{i=0}^{\infty} x_{i} y_{i}\) (Note that this makes sense since only finitely many \(x_{i}\) and \(y_{i}\) are nonzero.) Finally define $$ U=\left\\{\left[x_{i}\right) \text { in } V \mid \sum_{i=0}^{\infty} x_{i}=0\right\\} $$ a. Show that \(V\) is a vector space and that \(U\) is a subspace. b. Show that \(\langle,\rangle\) is an inner product on \(V\). c. Show that \(U^{\perp}=\\{\mathbf{0}\\}\). d. Hence show that \(U \oplus U^{\perp} \neq V\) and \(U \neq U^{\perp \perp}\).

Let \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be given by $$ T(a, b)=(b-a, a+2 b) $$ Show that \(T\) is symmetric if the dot product is used in \(\mathbb{R}^{2}\) but that it is not symmetric if the following inner product is used: $$ \langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x} A \mathbf{y}^{T}, A=\left[\begin{array}{rr} 1 & -1 \\ -1 & 2 \end{array}\right] $$

\(V\) denotes a finite dimensional inner product space. Define \(T: \mathbf{P} \rightarrow \mathbf{P}\) by \(T(f)=x f(x)\) 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 \(\langle f, g\rangle=a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}+\cdots\) a. Show that \(\langle,\rangle\) is an inner product on \(\mathbf{P}\). b. Show that \(T\) is an isometry of \(\mathbf{P}\). c. Show that \(T\) is one-to-one but not onto.
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