Problem 18 - Advanced Engineering Mathematics

Question

An inner product defined on the vector space \(P_{2}\) of all polynomials of degree less than or equal to 2 , is given by $$ (p, q)=\int_{-1}^{1} p(x) q(x) d x $$ Use the Gram-Schmidt orthogonalization process to transform the given basis \(B\) for \(P_{2}\) into an orthogonal basis \(B^{\prime}\). $$ B=\left\\{x^{2}-x, x^{2}+1,1-x^{2}\right\\} $$

Step-by-Step Solution

Answer

Use Gram-Schmidt to find orthogonal \( u_i \) and then normalize to get \( B' \).

Step 1: Compute the Gram-Schmidt Process for Basis Element 1

We start with the first basis vector, \( v_1 = x^2 - x \). Since \( v_1 \) is the first vector, it's automatically orthogonal to itself. We have \( u_1 = v_1 = x^2 - x \).

Step 2: Orthogonalize the Second Basis Element

Next, take the second basis vector \( v_2 = x^2 + 1 \) and make it orthogonal to \( u_1 \). The formula used is \( \text{proj}_{u_1}(v_2) = \frac{(v_2, u_1)}{(u_1, u_1)} u_1 \). Calculate these integrals:1. \( (v_2, u_1) = \int_{-1}^{1} (x^2 + 1)(x^2 - x) dx \)2. \( (u_1, u_1) = \int_{-1}^{1} (x^2 - x)^2 dx \)After computing, subtract the projection from \( v_2 \) to obtain \( u_2 = v_2 - \text{proj}_{u_1}(v_2) \).

Step 3: Orthogonalize the Third Basis Element

Proceeding to the third basis vector \( v_3 = 1 - x^2 \), make \( v_3 \) orthogonal to both \( u_1 \) and \( u_2 \). Apply Gram-Schmidt:1. Compute \( \text{proj}_{u_1}(v_3) = \frac{(v_3, u_1)}{(u_1, u_1)} u_1 \)2. Compute \( \text{proj}_{u_2}(v_3) = \frac{(v_3, u_2)}{(u_2, u_2)} u_2 \)Now calculate \( u_3 = v_3 - \text{proj}_{u_1}(v_3) - \text{proj}_{u_2}(v_3) \) to complete the orthogonalization.

Step 4: Normalize the Orthogonal Basis

Finally, normalize each of the orthogonal vectors \( u_1, u_2, \) and \( u_3 \) to form an orthonormal basis. Divide each vector by its norm calculated as \( \|u_i\| = \sqrt{(u_i, u_i)} \) for \( i = 1, 2, 3 \). This yields the orthonormal basis \( B' \).

Key Concepts

Inner Product Space

An inner product space is an abstract mathematical structure that extends the idea of the dot product to more complex, infinite-dimensional vector spaces. The core idea of an inner product space is that it allows us to define and calculate products between vectors, which provides a notion of angle and length.
In our exercise, the vector space we're looking at is the polynomial vector space, denoted by \( P_2 \). Here, the vectors are polynomials, and the specific inner product is defined as \( (p, q) = \int_{-1}^{1} p(x) q(x) \, dx \).
This operation isn't multiplication as we know it in basic arithmetic; it's an integration operation that combines two functions over a specified interval to generate a single number. This number can then be used to understand geometrical and analytical properties of shapes formed by these polynomials.

Orthogonal Basis

An orthogonal basis in a vector space is a set of basis vectors that are all orthogonal to each other. In simpler terms, each pair of different basis vectors is perpendicular.
Orthogonality is a useful property because it simplifies many calculations. For instance, when dealing with an orthogonal basis, the projection of a vector onto a subspace can be computed easily by considering single components one at a time.

• Orthogonality is confirmed when the inner product of the vectors equals zero, i.e., \( (v_i, v_j) = 0 \), where \( i eq j \).
• To obtain an orthogonal basis from a regular set of basis vectors, we use the Gram-Schmidt process.

By transforming a standard basis set into an orthogonal basis, we make vector operations easier and more computationally efficient.

Vector Space

A vector space is a fundamental concept in linear algebra and mathematics in general. It is a set equipped with two operations: vector addition and scalar multiplication. These operations must follow certain rules and axioms, such as commutativity and associativity.

• Vectors in a vector space can be added together to obtain another vector within the same space.
• Vectors can be multiplied by scalars, which are numbers from an associated field, to scale them.

The specific vector space involved in our exercise, \( P_2 \), consists of all polynomials with a degree of up to 2. This particular polynomial vector space forms an essential backdrop for understanding the inner product and the orthogonalization process.

Polynomial Vector Space

Polynomial vector spaces are special types of vector spaces where the elements, or vectors, are polynomial functions.
In our exercise, \( P_2 \) is the space of polynomials with degrees less than or equal to 2. This includes polynomials like \( x^2 - x \), \( x^2 + 1 \), and \( 1 - x^2 \).
They conform to all the axioms of vector spaces, meaning they can be added and multiplied by scalars while remaining within the space. They form a structured and rich space that lends itself well to orthogonalization approaches like the Gram-Schmidt process. This is because it allows us to break down complex polynomial forms into simpler, non-redundant components.