Let V be the space C[-1,1] with the inner product of Example 7 . Find an orthogonal basis for th

Let V be the space C[-1,1] with the inner product of Example...

Key Concepts

Polynomial Function Spaces

Polynomial function spaces consist of functions that can be expressed as polynomials and are a common setting for approximation techniques and orthogonalization. In these spaces, functions like constant, linear, and quadratic polynomials serve as simple yet fundamental building blocks for more complex function spaces and analyses.

Inner Product Spaces

An inner product space is a vector space equipped with an additional structure called an inner product, which allows for the measurement of angles and lengths within the space. In function spaces, the inner product is often defined as an integral over a specific interval, providing a way to generalize geometric concepts to functional analysis.

Legendre Polynomials

Legendre polynomials are a sequence of orthogonal polynomials that arise when applying the Gram-Schmidt process to the set of polynomials over the interval [-1, 1] with a specific inner product. They have important applications in physics and engineering, particularly in solving problems with spherical symmetry.

Orthogonal Basis

An orthogonal basis in a vector space is a set of vectors that are all mutually perpendicular to one another with respect to a given inner product. This property simplifies many aspects of linear algebra and function analysis since operations such as projections and decompositions become more straightforward when the basis vectors are orthogonal.

Gram-Schmidt Process

The Gram-Schmidt process is a method for converting a set of linearly independent vectors into an orthogonal (or orthonormal) set while maintaining the span of the original set. This technique is essential in many areas of mathematics as it provides a systematic way to generate an orthogonal basis for a vector space.

Transcript

0:00 Hello there.

00:02 Okay, for this exercise we are going to consider the space defined over the continuous functions on the interval minus 1 -1 with the inner product equipped when an inner product defined as the integral between minus 1 and 1 of the multiplication of the functions.

00:27 And in particular in this exercise we need to find some orthogonal basis, so we need to find an orthogonal basis, the span by the following polynomials, one, t and t squared.

00:52 Okay, so how to do this? well, we need to think of it, not as functions, but as vectors.

01:01 Okay, so we are going to use the grinchmidt procedure to obtain orthovenile basis.

01:05 So let's start to see what happened with these two polynomials here, one and t.

01:11 So they are autonomous, well let's see by computing the inner product.

01:16 So let us start by considering the inner product between 1 and t.

01:19 This is just the integral within minus 1 and 1 of t.

01:25 Sorry here is t.

01:29 T vt.

01:31 Okay.

01:33 So here we got this interval minus 1 -1, and t is an off function.

01:41 So this is equal to zero.

01:45 Or you can just integrate as the usual way and you will find that is equal to zero.

01:51 Okay, this is due to the symmetry.

01:56 You can think of this also geometrically.

02:00 T is just a straight line.

02:04 So the integral in this symmetric interval minus 1 -1, you can see that the area here and the area here are the so they are going to cancel between them and this is going to be zero.

02:21 At the end, okay? so this happened with other functions in general.

02:26 Or you can just integrate this and you will see that you will, let's make it explicitly, just in this case, this will be t squared divided 2, evaluates on the interval 1 minus 1.

02:39 This will be 1 half minus 1 half and this is equal to 0.

02:43 So this inner product is 0...

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