00:01
Hi here for the given question.
00:03
We need to calculate orthonormal basis of the subspace f1 comma f2 comma f3 in l square minus 1 to 1.
00:22
Now here we will use the gram -schmidt process in order to find the orthonormal basis.
00:27
So here in our case, we are given that f1 of x is equal to 1 f2 of x is equal to x and f3 of x is equal to x square.
00:38
So here in our case, we need to define the interval as minus 1 to 1.
00:45
This is our interval.
00:48
So here in our case, first of all, we will start with orthogonalization.
00:52
So here the first step will be orthogonalization.
01:02
This is the first step.
01:04
Now here we know that here we will define the first vector as v1 is equal to f1 which is equal to 1.
01:13
Now here for v2, it is a projection on v1.
01:17
So v2 is equal to f2 minus projection of v1 on f2.
01:22
So here this can be further written as the value of projection of v1 on f2 is equal to f2 v1 inner product of f2 and v1 divided by inner product of v1 and v1 multiplied with v1.
01:37
So here in our case, first of all, we will find the value of this projection.
01:42
So here the value of this projection can be written as here this is equal to integration over minus 1 to 1 x multiplied with 1 dx divided by integration over minus 1 to 1 and here we have 1 square dx.
02:02
So simplifying this and it is multiplied with 1.
02:05
So simplifying this we have value equals to 0.
02:08
So here v2 will be equal to f2 minus 0 which will be equal to x minus 0 which is equal to x.
02:16
Similarly v3 can be written as here in our case.
02:19
We have f3 minus projection of v1 f3 minus projection of v2 and here we have f3.
02:29
So here using the formula projection here in our case, we can find the value of projection of v1 or f3 which is equal to f3 v1 inner product of f3 v1 multiplied with v1 v1 divided by inner product of v1 v1 multiplied with v1...