00:01
In this question we have to find out the orthonormal basis for r3 with respect to the inner product space defined by this matrix which is 4 -2 0 -2 3 -1 0 -1 2.
00:16
So we will find out the orthonormal basis using gram schmidt process.
00:24
So according to this process first of all we need the three vectors.
00:28
So three vectors will be u1 u2 u3.
00:33
U1 will be equal to 4 -2 0.
00:36
U2 will be equal to minus 2 3 minus 1 and u3 will be equal to 0 minus 1 2.
00:44
Now according to this process first of all we will let v1 is equal to u1 which is 4 -2 0.
00:53
V2 will be equal to u2 minus inner product space of u2 v1 divided by norm of v1 square into v1.
01:07
Now here let us find out the inner product space of u2 and v1.
01:12
So u2 is minus 2 3 minus 1 and v1 is 4 -2 0.
01:19
So we will multiply them component wise.
01:21
So this is minus 8.
01:23
This will be minus 6 and this is 0.
01:25
So this is minus 14.
01:28
And if we find out norm of v1 square.
01:33
So this will be square root of v1 is this.
01:38
So 4 square is 16 minus 2 square is 4 plus 0 and square of this.
01:43
So square will be cancelled by the square root.
01:45
So this will be equal to 20.
01:47
Now putting all of the values.
01:48
So this is minus 2 3 minus 1 minus minus 14 by 20 and v1 is 4 minus 2 0.
01:58
So this is 2 7's are 2 10's are.
02:02
So this will be minus 2 3 minus 1 minus minus plus and this is 28 by 10 minus 14 by 10 and this is 0.
02:12
So on solving this we will get that v2 is equal to 8 by 10 16 by 10 and this is minus 1.
02:22
So this is the value of v2...