Problem 20 - Differential Equations and Linear Algebra

Question

Determine an orthogonal basis for the subspace of \(C^{0}[a, b]\) spanned by the given vectors, for the given interval \([a, b] .\) Use the inner product given in Equation \((5.1 .5)\). $$\begin{array}{l} f_{1}(x)=1, \quad f_{2}(x)=\sin x, \quad f_{3}(x)=\cos x \\ a=-\pi / 2, \quad b=\pi / 2 \end{array}$$

Step-by-Step Solution

Answer

The orthogonal basis for the subspace of \(C^{0}[a, b]\) spanned by the given vectors, for the given interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\), using the Gram-Schmidt process with the given inner product, is composed of the functions: \(u_1(x) = 1\), \(u_2(x)=\sin{x}\), and \(u_3(x)=\cos{x}\).

Step 1: Analyze the problem

Identify the type of problem and the appropriate mathematical technique to apply.

Step 2: Apply the technique and solve

The orthogonal basis for the subspace of \(C^{0}[a, b]\) spanned by the given vectors, for the given interval \([-\frac{.

Step 3: Verify the result

Check the answer by substitution or alternative methods to confirm correctness.

Key Concepts

Gram-Schmidt Process

The Gram-Schmidt process is a method used in linear algebra to orthogonalize a set of vectors in an inner product space. It's an essential tool for converting any set of linearly independent vectors into an orthogonal set. Orthogonal vectors have the benefit of simplifying many calculations, including projections and transformation matrices.

To start the Gram-Schmidt Process, you take a set of linearly independent vectors, for instance, the functions \(f_1(x) = 1\), \(f_2(x) = \sin x\), and \(f_3(x) = \cos x\). Then, you create a new set of orthogonal vectors \(u_1(x)\), \(u_2(x)\), and \(u_3(x)\).

• Start with: Set the first orthogonal vector \(u_1\) as equal to \(f_1\).
• Next vectors: Subtract projections from successive vectors to maintain orthogonality.

This method ensures that the resulting vectors \(u_1, u_2, u_3\) not only span the same subspace but also are orthogonal to each other.

Inner Product

An inner product is a generalization of the dot product of vectors to more abstract spaces, which allows measuring lengths and angles between vectors. It's a fundamental concept in linear algebra, particularly when dealing with function spaces.

For the functions \(f_1(x) = 1\), \(f_2(x) = \sin x\), and \(f_3(x) = \cos x\), an example of a suitable inner product is:\[\langle f, g \rangle = \int_{a}^{b} f(x) g(x) \, dx\]where \([a, b]\) is the given interval.

• Calculating Inner Products: For instance, \(\langle f_2, u_1 \rangle = \int_{-\pi/2}^{\pi/2} \sin x \cdot 1 \, dx = 0\), indicating orthogonality.
• Using Inner Products: These values are used in the Gram-Schmidt process to ensure orthogonality of the constructed basis.

Remember, the inner product provides insights about the function space structure and is leveraged to apply the Gram-Schmidt Process accurately.

Linear Algebra

Linear Algebra is the branch of mathematics that deals with vectors and vector spaces. It includes studying lines, planes, and subspaces, understanding transformations, and analyzing various vector operations.

In the problem at hand, Linear Algebra concepts are used to transform functions into an orthogonal basis in the space \(C^{0}[a, b]\). Specifically, linear combinations are foundational, allowing you to form vectors (or functions) from given sets.

• Subspaces: The set of all linear combinations forms a subspace of \(C^{0}[a, b]\).
• Basis: A set of vectors (or functions) that span a subspace are called a basis. Using the Gram-Schmidt process, we convert this basis into an orthogonal one.

Linear Algebra provides the tools needed for such transformations and ensures that these transformations maintain the structure and properties of vector spaces.

Function Space

Function spaces, like \(C^{0}[a, b]\), consist of all continuous functions defined over a specified interval. These spaces can be explored using concepts from linear algebra such as vector spaces, bases, and linear transformations.

In this context, \(f_1(x)\), \(f_2(x)\), and \(f_3(x)\) are continuous functions on the interval \([-\pi/2, \pi/2]\), and together they span a subspace within the larger function space.

• Continuous Functions: These functions don't have breaks, making integration and inner product calculations feasible.
• Spanning a Subspace: Functions can be manipulated using Linear Algebra techniques to span relevant subspaces, allowing for tasks like orthogonalization.

By using a function space framework, we can apply linear algebra concepts effectively, such as constructing orthogonal bases through the Gram-Schmidt process.