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Chapter 3: Problem 26

Prove that the vectors \(\|v\| w+\|w\| v\) and \(\|v\| w-\|w\| v\) in an inner- product space \(V\) are perpendicular.

Short Answer

Expert verified
Vectors \(\|v\| w+\|w\| v\) and \(\|v\| w-\|w\| v\) are perpendicular because their inner product is zero.

Step by step solution

01

Understand Vector Notation and Definitions

The vectors given are \(\|v\| w+\|w\| v\) and \(\|v\| w-\|w\| v\), where \(v\) and \(w\) are vectors in an inner-product space \(V\). The notation \(\|v\|\) represents the norm of the vector, which is defined as \(\sqrt{\langle v, v \rangle}\), where \(\langle v, v \rangle\) is the inner product of \(v\) with itself.
02

Define Perpendicular Vectors in Inner-product Spaces

Two vectors \(a\) and \(b\) in an inner-product space \(V\) are said to be perpendicular (or orthogonal) if their inner product equals zero: \(\langle a, b \rangle = 0\). We need to show that \(\langle \|v\| w+\|w\| v, \|v\| w-\|w\| v \rangle = 0\).
03

Calculate the Inner Product

Compute the inner product of the two vectors:\[langle \|v\| w+\|w\| v, \|v\| w-\|w\| v \rangle = \langle \|v\| w, \|v\| w \rangle - \langle \|v\| w, \|w\| v \rangle + \langle \|w\| v, \|v\| w \rangle - \langle \|w\| v, \|w\| v \rangle\]Due to properties of inner products, the second and third terms simplify to 0.
04

Simplify Using Inner-product Properties

Apply the linearity and symmetry properties of the inner product:1. \(\langle \|v\| w, \|v\| w \rangle = \|v\|^2 \langle w, w \rangle \)2. \(\langle \|v\| w, \|w\| v \rangle = \|v\|\|w\| \langle w, v \rangle \)3. \(\langle \|w\| v, \|v\| w \rangle = \|v\|\|w\| \langle v, w \rangle \ = \langle \|v\| w, \|w\| v \rangle\)4. \(\langle \|w\| v, \|w\| v \rangle = \|w\|^2 \langle v, v \rangle \)Noticing the symmetry, the negative terms cancel with the positive terms.
05

Confirm Orthogonality

Given the simplifications:\[\langle \|v\| w+\|w\| v, \|v\| w-\|w\| v \rangle = \|v\|^2 \langle w, w \rangle - \|w\|^2 \langle v, v \rangle = 0\]This means the vectors are orthogonal, thus confirming \(\|v\| w+\|w\| v\) and \(\|v\| w-\|w\| v\) are perpendicular.

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Key Concepts

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

Perpendicular Vectors
Perpendicular vectors, also known as orthogonal vectors, play a pivotal role in inner-product spaces. When two vectors are perpendicular, their inner product is zero. This concept, \(\langle a, b \rangle = 0\), is crucial for understanding many areas of linear algebra and geometry.
  • Geometric Interpretation: Imagine two vectors as arrows meeting at a right angle; they form an 'L' shape. Perpendicular vectors extend this idea into multiple dimensions.
  • Application: In physical space, perpendicular vectors are common in force decompositions and changes of base.
Perpendicular vectors help in simplifying problems and determining vector spaces' orthogonality. Being perpendicular is a specific condition that can indicate important properties about the vectors, like independence in some contexts.
Vector Norms
The norm of a vector is a measure of its length or magnitude in an inner-product space. It gives a sense of how big a vector is. For any vector \(v\), the norm is represented as \(|v| = \sqrt{\langle v, v \rangle}\). This formula derives from the inner product of a vector with itself.
  • Intuition: Consider the norm as the "size" of the vector. If you imagine \(v\) as a directed line segment from the origin, \(|v|\) is its geometrical length.
  • Properties: A vector's norm is always a non-negative scalar. Importantly, if \(|v| = 0\), the vector is the zero vector, which means it has no direction or length.
  • Applications: Vector norms are widely used in scaling vectors to unit length, which is crucial for normalization processes in mathematics and data science.
Understanding norms helps in measuring distances and angles in vector spaces, critical when assessing vector relationships such as perpendicularity and orthogonality.
Orthogonality
Orthogonality is a broad concept stemming from perpendicularity that applies to vectors more generally. Two vectors in an inner-product space are orthogonal when their inner product is zero. This concept extends beyond simple right-angle relationships.
  • Importance: Orthogonal vectors often represent independent directions in vector space, making them fundamental in algorithms like Gram-Schmidt and techniques like singular value decomposition.
  • Orthogonal Sets: A set of vectors is orthogonal if each pair of vectors in the set is orthogonal. This means no vector in the set can be written as a linear combination of others, a useful property for constructing orthonormal bases.
  • Practical Use: Orthogonal vectors simplify many calculations, showing their importance in simplifying linear transformations and solving equations.
Orthogonality provides insights into the structure and simplicity of vector spaces, aiding in processes like simplifying matrix computations and offering a foundation for understanding linear independence.
Inner Product Properties
Inner-product spaces are defined by their inner product, which has specific properties. Understanding these properties can simplify proving vector relationships such as perpendicularity, among others.
  • Linearity: The inner product is linear in its first and second arguments separately. This means functions like \(\langle au + bv, w \rangle = a \langle u, w \rangle + b \langle v, w \rangle\).
  • Symmetry: For any vectors \(v\) and \(w\), \(\langle v, w \rangle = \langle w, v \rangle\). This property is particularly meaningful in simplifying expressions where swapping terms might reduce complexity.
  • Positive Definite: The inner product of a vector with itself is always non-negative, \(\langle v, v \rangle \geq 0\), and \(\langle v, v \rangle = 0\) if and only if \(v\) is the zero vector.
Understanding these properties helps in effectively applying and manipulating inner-product spaces, aiding in identifying orthogonality and other relationships. These properties are foundational for working with complex vector spaces across mathematics and engineering.

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