Find a polynomial p3 such that {p0, p1, p2, p3} (see Exercise 11) is an orthogonal basis for the

Find a polynomial p3 such that {p0, p1, p2, p3} (see...

Key Concepts

Polynomial Spaces

A polynomial space, often denoted by P_n, is the collection of all polynomials of a degree up to n. This space forms a vector space under the usual operations of polynomial addition and scalar multiplication. Understanding polynomial spaces is crucial because they provide a structured environment where concepts like basis, dimension, and inner products can be applied.

Orthogonal Basis

An orthogonal basis is a set of basis vectors in which every pair of distinct vectors is orthogonal with respect to an inner product defined on the space. In polynomial spaces, constructing an orthogonal basis (often consisting of orthogonal polynomials) simplifies many problems, including approximations and projections, because the inner product between any two different basis polynomials is zero.

Gram-Schmidt Process

The Gram-Schmidt process is a method for converting a set of linearly independent vectors (or functions) into an orthogonal (or orthonormal) set. In the context of polynomial spaces, this process is used to generate an orthogonal basis by systematically subtracting the projections onto already established orthogonal polynomials. It is a fundamental tool for ensuring the orthogonality of the basis within the space.

Scaling of Polynomials

Scaling a polynomial involves multiplying it by a non-zero constant factor so that the polynomial meets a specific criterion or normalization condition. This is often done to achieve a particular leading coefficient, norm, or value vector, allowing the polynomial to exhibit desired properties in its evaluation or representation.

Vector Representation of Polynomials

Polynomials can be represented as vectors by evaluating them at selected points. This vector of values encapsulates the information of the polynomial in a finite-dimensional form, thereby facilitating operations like comparison, orthogonalization, and scaling in terms of standard vector space techniques.

Transcript

00:01 We have the orthogonal basis p0, p1, p2, given by the polynomials 1, t, and t squared minus 2.

00:09 We know this from example 5.

00:11 So let me write p0 bar for the vector obtained by computing p0 in minus 2, in minus 1, in 0, in 1, and in 2.

00:23 So this is just the vector of 1s.

00:26 And then p1 bar is minus 2, minus 1, 0 and 2.

00:30 And finally p2 bar is 2 minus 1 minus 2 minus 1 and 2 now we write these vectors because we are called the inner product between polynomials so in this case for instance p 0 against b 0 is given by p 0 bar dot p 0 bar so in this case is 5 and similarly we compute p 1 against p 1 to be 10 and p2 against b2 to be 14 we compute these numbers now because we are going to use them now, well soon, when we are doing the grand schmitt process.

01:07 So let's consider the polynomial q, which is given by t cubed.

01:12 Let's write q bar.

01:13 Of course, this is minus 8, minus 1, 0, 1 and 8.

01:17 And we want to apply the grans schmidt process to q to obtain, well, to obtain a vector which is orthogonal to both p0, p1 and p2.

01:28 So now the computation is simplified by the fact that the vectors p0, p1 and p2 are, well, the vectors, the polynomials, p0, p1, p2 are already orthogonal.

01:40 So all we need to do is to compute the projection of q onto p0, p1 and p2.

01:49 So let's start with p0, which is q against p0, divided by p0 against p0.

01:55 And of course we recall that this is q bar dot p0 bar divided by 5 the 5 we computed previously and then p 0 is just the constant function 1 and now you can compute a q bar dot p0 bar is 0...

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