Question

\(V\) denotes a finite dimensional inner product space. Exercise \(\mathbf{1 0 . 4 . 1}\) Show that the following linear operators are isometries. $$ \text { a. } T: \mathbb{C} \rightarrow \mathbb{C} ; T(z)=\bar{z} ;\langle z, w\rangle=\operatorname{re}(z \bar{w}) $$ b. \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} ; T\left(a_{1}, a_{2}, \ldots, a_{n}\right)\) \(\quad=\left(a_{n}, a_{n-1}, \ldots, a_{2}, a_{1}\right) ;\) dot product c. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} ; \quad T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{ll}c & d \\ b & a\end{array}\right]\) \(\quad\langle A, B\rangle=\operatorname{tr}\left(A B^{T}\right)\) d. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} ; T(a, b, c)=\frac{1}{9}(2 a+2 b-c, 2 a+\) \(2 c-b, 2 b+2 c-a) ;\) dot product

Short answer

All four transformations are isometries as they preserve the inner product.

Step-by-step solution

Understand the Definition of Isometry

An isometry is a linear operator that preserves the inner product. For a linear operator \(T\) to be an isometry, it must satisfy \( \langle T(x), T(y) \rangle = \langle x, y \rangle \) for all vectors \(x\) and \(y\) in the space.

Analyze Part (a)

For part (a), we need to show that \( T(z) = \bar{z} \) is an isometry. The inner product is \( \langle z, w \rangle = \operatorname{Re}(z \bar{w}) \). Check that \( \langle T(z), T(w) \rangle = \operatorname{Re}(\bar{z} w) = \operatorname{Re}(z \bar{w}) = \langle z, w \rangle \), confirming that \(T\) is an isometry.

Analyze Part (b)

For part (b), \( T(a_1, a_2, \ldots, a_n) = (a_n, a_{n-1}, \ldots, a_1) \). The dot product is used as the inner product. We need to check that the Euclidean norm is preserved. Since rewriting or reversing the order of the elements doesn't change their magnitudes or the dot product, \( T \) is an isometry.

Analyze Part (c)

For part (c), \( T \) maps \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) to \( \begin{pmatrix} c & d \ b & a \end{pmatrix} \). The inner product is defined as \( \langle A, B \rangle = \operatorname{tr}(A B^T) \). Calculate that \( \operatorname{tr}(AB^T) = cd + ab \) after transposition keeps the trace unchanged, confirming \( T \) as an isometry.

Analyze Part (d)

In part (d), for \( T(a, b, c) = \frac{1}{9}(2a+2b-c, 2a+2c-b, 2b+2c-a) \), check if \( T \) preserves the dot product \( \langle x, y \rangle = a_1a_2 + b_1b_2 + c_1c_2 \). Calculate both sides of the condition for random vectors to confirm that \( T \) is indeed an isometry.

Key concepts

Inner Product Space

An inner product space is a vector space equipped with an additional structure called an inner product. This inner product allows us to measure angles and distances between vectors. Think of it like assigning a 'geometric flavor' to the space that might initially seem abstract.

The inner product can take various forms, including the dot product for real vectors or the Hermitian product for complex vectors. It fulfills several essential properties:

• Linearity in the first argument
• Symmetry, meaning the order doesn't affect the outcome for real numbers
• Positivity, ensuring non-negative results, with zero only when the vector itself is zero

Understanding the inner product is fundamental in Linear Algebra, as it is the cornerstone for defining lengths (norms) and angles between vectors. It helps in determining whether two vectors are orthogonal, akin to them forming a right angle in regular geometry.

Isometries

An isometry is a special type of linear transformation that preserves distances within an inner product space. When working with isometries, the critical idea is that the transformation should not distort the space's structure. For instance, if two vectors form a specific angle before an isometry, they maintain that angle afterward.

For a linear operator \( T \) to be considered an isometry, it must preserve the inner product: \( \langle T(x), T(y) \rangle = \langle x, y \rangle \) for all vectors \( x \) and \( y \). This isometrical property ensures that lengths and angles are maintained. It's like a perfect transformation without any stretching or squishing of the space.

This concept is pivotal in areas like computer graphics where preserving shape integrity is crucial. Isometries ensure that transformed objects remain congruent to their originals, maintaining their intrinsic properties.

Linear Operators

Lines and transformations are at the heart of linear algebra, and a linear operator is a function that maps one vector space to another while respecting vector addition and scalar multiplication. This means if you scale or add vectors, then transform them, it's the same as transforming the vectors individually and then adding or scaling them.

Linear operators are foundational in understanding transformations in vector spaces. They provide a framework through which we can describe rotations, reflections, and other geometric transformations mathematically.

• For example, a reflection operator might take a point over a line or plane, altering its orientation but not its distance from the origin.

With linear operators, complex transformations can be broken down into simpler components, making them easier to analyze, especially when combined with other operators. Understanding these helps in areas ranging from quantum mechanics to computer graphics, where detecting edges and shapes involve linear transformations.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector mathematics. It takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.

The dot product of vectors \( \mathbf{a} = (a_1, a_2, \, ..., a_n) \) and \( \mathbf{b} = (b_1, b_2, \, ..., b_n) \) is calculated as:\[ a_1b_1 + a_2b_2 + \cdots + a_nb_n \]
This operation results in a scalar and has significant implications:

• Magnitude and direction: The dot product relates to the cosine of the angle between the vectors and can show whether vectors are orthogonal (produces zero).
• Projection: It is used to determine the component of one vector along another.

The dot product is widely used because it links algebraic and geometric ideas, allowing problems in geometry to be reformulated and solved using algebraic methods. It also plays a crucial role in defining key concepts like orthogonality and angles in higher-dimensional spaces.