00:01
All right, so we need to check symmetry.
00:06
So let's go ahead and do this.
00:08
This is the same thing as t of x, t of y.
00:14
Now, this original inner product is symmetric, so i can swap them.
00:21
And then this is yx prime.
00:25
So it's symmetric.
00:26
Next, we need to check linearity.
00:28
So i'll go ahead and do ax plus y, z.
00:33
This is t of, so prime here, this is t of, a x plus y t of z with the original inner product t is linear so this is a t of x plus t of y paired with t of z and then my original inner product is linear so i can write this as a t of x t of z plus t of y t of z and this is exactly what we wanted this is a times x z plus y, z.
01:11
You can show linearity in the other argument, the second entry, using the exact same argument, so i'll skip that part.
01:19
Finally, we need to show that if i take x, comma, x, i need this to be greater than or equal to zero, and zero if and only if x equals zero...