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Chapter 4: Problem 279

Check that the vector space \(\mathbf{R}^{k}\), equipped with the imer product \(\left(x_{1}, \ldots, x_{k}\right) \cdot\left(y_{1} \ldots, y_{k}\right)=x_{1} y_{1}+\cdots+x_{k} y_{k}\), is Euclidean, and compute lengths and pairwise angles of the \(k\) vectors \((1.0, \ldots, 0)\). \(\ldots,(0, \ldots, 0,1)\).

Short Answer

Expert verified
\( \mathbf{R}^k \) is Euclidean. Each vector has length 1, and angles between distinct vectors are \( 90^\circ \).

Step by step solution

01

Definition of Inner Product Space

In the vector space \( \mathbf{R}^{k} \), the inner product between two vectors \( \mathbf{x} = (x_1, x_2, \ldots, x_k) \) and \( \mathbf{y} = (y_1, y_2, \ldots, y_k) \) is defined as \( \mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2 + \cdots + x_ky_k \). We need to verify if this inner product satisfies the properties of a Euclidean space.
02

Verify Euclidean Space Properties

The inner product must satisfy the following properties for \( \mathbf{R}^k \) to be Euclidean: 1) **Commutativity:** \( \mathbf{x} \cdot \mathbf{y} = \mathbf{y} \cdot \mathbf{x} \). 2) **Distributivity:** \( \mathbf{x} \cdot (\mathbf{y} + \mathbf{z}) = \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z} \). 3) **Scalar Multiplication:** \( (c\mathbf{x}) \cdot \mathbf{y} = c(\mathbf{x} \cdot \mathbf{y}) \). 4) **Positive Definiteness:** \( \mathbf{x} \cdot \mathbf{x} \geq 0 \) and \( \mathbf{x} \cdot \mathbf{x} = 0 \) implies \( \mathbf{x} = \mathbf{0} \). All these properties hold for the defined inner product in \( \mathbf{R}^k \).
03

Compute Vector Lengths

The length (or norm) of a vector \( \mathbf{x} = (x_1, x_2, \ldots, x_k) \) in an inner product space is given by \( \|\mathbf{x}\| = \sqrt{\mathbf{x} \cdot \mathbf{x}} \). For the standard basis vectors \( (1,0,\ldots,0), (0,1,\ldots,0), \ldots, (0,0,\ldots,1) \), the length of each is \( \|(1,0,\ldots,0)\| = \sqrt{1^2} = 1 \), and similarly for the others, since each vector contains a single 1 and all other components are 0.
04

Compute Pairwise Angles Between Vectors

The angle \( \theta \) between two vectors \( \mathbf{x} \) and \( \mathbf{y} \) is given by \( \cos \theta = \frac{\mathbf{x} \cdot \mathbf{y}}{\|\mathbf{x}\| \|\mathbf{y}\|} \). For any two distinct standard basis vectors, like \( (1,0,\ldots,0) \) and \( (0,1,\ldots,0) \), the inner product is 0, resulting in \( \cos \theta = 0 \), which corresponds to \( \theta = 90^\circ \). This holds for any pair of different basis vectors, confirming they are orthogonal.

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Key Concepts

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

Inner Product
The inner product is an essential operation in a Euclidean vector space, helping us measure the similarity or perpendicularity of vectors. In the vector space \( \mathbf{R}^{k} \), the inner product between two vectors \( \mathbf{x} = (x_1, x_2, \ldots, x_k) \) and \( \mathbf{y} = (y_1, y_2, \ldots, y_k) \) is calculated as:
\[ \mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2 + \cdots + x_ky_k. \]
This operation fulfills key properties that ensure \( \mathbf{R}^{k} \) qualifies as a Euclidean space:
  • Commutativity: The sequence of the vectors doesn't matter; \( \mathbf{x} \cdot \mathbf{y} = \mathbf{y} \cdot \mathbf{x} \).
  • Distributivity: Your vectors can be distributed across an addition; \( \mathbf{x} \cdot (\mathbf{y} + \mathbf{z}) = \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z} \).
  • Scalar Multiplication: The product is scalable; \( (c\mathbf{x}) \cdot \mathbf{y} = c(\mathbf{x} \cdot \mathbf{y}) \).
  • Positive Definiteness: Each vector generates a non-negative value with itself; \( \mathbf{x} \cdot \mathbf{x} \geq 0 \) and is zero only if \( \mathbf{x} \) is the zero vector.
Vector Length
The length, or norm, of a vector is a measure of its size. In our context, it helps us understand the magnitude of different vectors within the Euclidean space. We calculate the length of a vector \( \mathbf{x} = (x_1, x_2, \ldots, x_k) \) using the formula:
\[ \|\mathbf{x}\| = \sqrt{\mathbf{x} \cdot \mathbf{x}} \]
For the standard basis vectors like \( (1,0,\ldots,0) \), we apply the formula:
  • The inner product \( \mathbf{x} \cdot \mathbf{x} \) results in \( 1 \), since only one component is non-zero and equal to 1.
  • Thus, their length is \( \|\mathbf{x}\| = \sqrt{1} = 1 \).
The simplicity of these computations reinforces why standard basis vectors are crucial components of any vector space.
Basis Vectors
Basis vectors offer a foundation for building any vector in the vector space. They serve as building blocks, offering coordinates from which we compose other vectors. In \( \mathbf{R}^{k} \), these are the standard basis vectors:\[(1,0,\ldots,0), (0,1,\ldots,0), \ldots, (0,0,\ldots,1)\]
These vectors have a special status because:
  • Each one has a magnitude of 1, ensuring that they can scale without altering the space's structure.
  • They are orthogonal to each other, meaning their inner product with one another is zero.
This orthogonality ensures that we can easily measure angles and distances between them 鈥� a cornerstone of vector operations in Euclidean spaces.
Angle Between Vectors
The angle between vectors is a crucial concept that reveals how two vectors are oriented with respect to each other. Calculating the angle \( \theta \) involves the cosine of the angle, given by the formula:
\[ \cos \theta = \frac{\mathbf{x} \cdot \mathbf{y}}{\|\mathbf{x}\| \|\mathbf{y}\|} \]
For standard basis vectors such as \( (1,0,\ldots,0) \) and \( (0,1,\ldots,0) \), the inner product is zero:
  • This means \( \cos \theta = 0 \), which simplifies to an angle of \( \theta = 90^\circ \).
  • Thus, all standard basis vectors are orthogonal, strengthening their role in properly structuring the vector space.
Understanding angles between vectors assists in visualizing the geometric relationships between different directions in a vector space. This concept is pivotal when analyzing shapes, movements, and rotations within the space.

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Most popular questions from this chapter

Prove the triangle inequality: for arbitrary radius-vectors \(\vec{u}, \vec{v} . \vec{w}\) in a Euclidean vector space, \(|\vec{u}-\vec{v}|+|\vec{v}-\vec{u}| \geq|\vec{u}-\vec{w}|\). Hint: Use the Cauchy Schwarz inequality.

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