Problem 31 - Differential Equations and Linear Algebra

Question

Use the previous exercise together with the inner product space axioms to derive a formula for \(\langle\mathbf{u}+\mathbf{v}, \mathbf{w}+\mathbf{x}\rangle\) for vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{x}\) in an inner product space \(V\)

Step-by-Step Solution

Answer

The formula for the inner product of two vectors given as the sum of two other vectors in an inner product space V is: \(\langle\mathbf{u}+\mathbf{v},\mathbf{w}+\mathbf{x}\rangle = \langle\mathbf{u},\mathbf{w}\rangle + \langle\mathbf{v},\mathbf{w}\rangle + \langle\mathbf{u},\mathbf{x}\rangle + \langle\mathbf{v},\mathbf{x}\rangle\).

Step 1: Apply Distributivity on the Inner Product

First, apply the distributivity axiom on the given inner product: \(\langle\mathbf{u}+\mathbf{v},\mathbf{w}+\mathbf{x}\rangle = \langle\mathbf{u}+\mathbf{v},\mathbf{w}\rangle + \langle\mathbf{u}+\mathbf{v},\mathbf{x}\rangle\)

Step 2: Apply Distributivity Again on Each Term

Apply the distributivity axiom again on each term in the previous equation: \(\langle\mathbf{u}+\mathbf{v},\mathbf{w}\rangle + \langle\mathbf{u}+\mathbf{v},\mathbf{x}\rangle = (\langle\mathbf{u},\mathbf{w}\rangle + \langle\mathbf{v},\mathbf{w}\rangle) + (\langle\mathbf{u},\mathbf{x}\rangle + \langle\mathbf{v},\mathbf{x}\rangle)\)

Step 3: Combine the Terms

Combine all the terms together: \(\langle\mathbf{u}+\mathbf{v},\mathbf{w}+\mathbf{x}\rangle = \langle\mathbf{u},\mathbf{w}\rangle + \langle\mathbf{v},\mathbf{w}\rangle + \langle\mathbf{u},\mathbf{x}\rangle + \langle\mathbf{v},\mathbf{x}\rangle\) Thus, the formula for the inner product of two vectors given as the sum of two other vectors in an inner product space V is: \(\langle\mathbf{u}+\mathbf{v}, \mathbf{w}+\mathbf{x}\rangle = \langle\mathbf{u},\mathbf{w}\rangle + \langle\mathbf{v},\mathbf{w}\rangle + \langle\mathbf{u},\mathbf{x}\rangle + \langle\mathbf{v},\mathbf{x}\rangle\)

Key Concepts

Distributivity in Inner Product Spaces

In the vast world of linear algebra, inner product spaces are a crucial concept where vectors meet geometry. One of the key properties of inner product spaces is distributivity. Distributivity plays a vital role when dealing with vector operations in these spaces.

The distributive property allows us to decompose a complex inner product into simpler, more manageable terms. Consider four vectors in an inner product space:

• \(\mathbf{u}\) and \(\mathbf{v}\)
• \(\mathbf{w}\) and \(\mathbf{x}\)

The vector sum \(\langle \mathbf{u} + \mathbf{v}, \mathbf{w} + \mathbf{x} \rangle\) can be expanded using distributivity. Initially, we distribute the inner product over the addition of vectors: \(\langle \mathbf{u} + \mathbf{v}, \mathbf{w} + \mathbf{x} \rangle = \langle \mathbf{u} + \mathbf{v}, \mathbf{w} \rangle + \langle \mathbf{u} + \mathbf{v}, \mathbf{x} \rangle\). Further application of distributivity on each term ultimately simplifies to: \(\langle \mathbf{u}, \mathbf{w} \rangle + \langle \mathbf{v}, \mathbf{w} \rangle + \langle \mathbf{u}, \mathbf{x} \rangle + \langle \mathbf{v}, \mathbf{x} \rangle\).

This expression shows how distributivity allows complex operations to be broken down into simple, pairwise inner products. Understanding this expansion helps in tackling problems involving multiple vectors in inner product spaces efficiently.

Vector Operations

Vector operations form the backbone of numerous applications in mathematics and physics. Understanding these operations within inner product spaces is essential. In the context of the given exercise, we specifically deal with addition and inner products of vectors.

Addition of vectors, like \(\mathbf{u} + \mathbf{v}\) or \(\mathbf{w} + \mathbf{x}\), combines two vectors to produce a new vector. The resulting vector has components that are the sum of the corresponding components of the original vectors.

Inner product, on the other hand, is a more nuanced operation. It allows you to calculate a scalar from two vectors. This scalar gives insights into the relationship between these vectors, including their angle and length. For vectors \(\mathbf{a}\) and \(\mathbf{b}\), the inner product is denoted \(\langle \mathbf{a}, \mathbf{b} \rangle\).

When combined with distributive property, as seen in this exercise, these vector operations allow for the breaking down of complex expressions. As you sum and distribute operations, comprehension of vector addition and scalar generation through inner products becomes fundamental.

Linear Algebra Concepts

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces. It's vital in understanding systems of linear equations, vector operations, and transformations.

In the exercise at hand, we explored how inner product spaces leverage the principles of linear algebra. Key practices such as distributivity, linear combinations, and the inner product are paramount.

• Distributivity in linear algebra: Helps in breaking down complex expressions involving vector operations.
• Linear combinations: When adding vectors such as \(\mathbf{u} + \mathbf{v}\), we are essentially forming a linear combination, which is foundational in creating span and basis for vector spaces.
• Inner products: Provide a geometric interpretation of vectors, offering insight into angles and lengths, crucial for understanding vector spaces and their dimensions.

Understanding and mastering these concepts offers robust tools for analyzing and solving problems in various fields like engineering, data science, and more. Each concept builds on previous knowledge, creating an integrated understanding of linear spaces and operations within them.