The first four hermite polynomials are 1 2t 2 c 4t2 and 12t...

. The first four Hermite polynomials are 1, 2t, 2 C 4t2, and 12t C 8t3. These polynomials arise naturally in the study of certain important differential equations in mathematical physics.2 Show that the first four Hermite polynomials form a basis of P3.

Transcript

0:00 Hello there.

00:02 For the following exercise we have this we have two parts.

00:07 First we have this set of polynomials that actually corresponds to the remit polynomials and we need to show that this set v corresponds to a basis for this the vector space b3 that corresponds to the space of polynomials of at most degree 3.

00:26 Okay so the first step as i mentioned corresponds to that this is a basis.

00:32 And then we need to pick a particular polynomial and write that in the basis with respect to the basis b.

00:42 Okay so it's important to use the definition of a basis.

00:46 What is the meaning to be a basis? okay so a set of pointomials, a set of elements, a set of vectors in for a vector space form a basis if the first they are linearly independent and second if they expand the space.

01:08 In this case, p3.

01:10 These two conditions can be simplified as a system as a linear system of equations.

01:18 Okay? so the linearly independence on one side corresponds to pick these vectors a linear combination of these vectors, c1 plus it to separate them.

02:05 Okay, so when need to pick a linear a linear combination of these pointomials and they should be equals to the zero vector in that vector space and in this case corresponds to the zero point of this is the meaning of well this should be equals to to the zero point omel if and only if c1 it's equals to c2 is equals to c3 it's equals to c4 which are all them are equals to zero okay so the only is solution for this coefficient c1 up to c4 is that all of them are zero.

02:48 Okay, so that's the meaning of linear independence.

02:53 And this pan can be encoded to pick any vector and any pointomial in p3 where this pointomial will be equals to b1 plus b2 plus b3 b2 t plus b2 t plus b3 t squared plus b4 t cube so this polynomial can be written as a linear combination of the basis elements that's the meaning of the span so technically we have here a similar system but here instead of zero we have the polynomial or polynomial of interest b2 t plus b3 t squared plus b4 t cube okay we have these two systems we have these two equations but these can be represented as a set of linear equations and in the case of the span we need to to ask for a unique solution a unique solution for the for the constant c1 c2 c3 and c4 okay no so how to construct the system well basically let's focus on the span equation and we need to group all the constants terms okay all the terms that are not multiplying by any t so for for that from that we obtain c1 plus two times c3 equals to zero to b1 sorry to be 1.

04:59 Then we need to group all the terms that are multiplied by t...

Question

. The first four Hermite polynomials are 1, 2t, 2 C 4t2, and 12t C 8t3. These polynomials arise naturally in the study of certain important differential equations in mathematical physics.2 Show that the first four Hermite polynomials form a basis of P3.

Please give Ace some feedback