Hey I have having trouble with this problem:



I feel like Gram-Schmidt is the way to go here but not sure where to start. I don't think I can just start with an arbitrary basis for V and extend it using Gram Schmidt and then show that the resulting basis is orthogonal.

Can anyone help? Thanks!!

Your on the right track in using that method. Indeed all finite vector spaces have orthogonal basis. You can look at the following link and go to orthonormal Sequences to see how to go about starting with a random basis and moving into a orthogonal basis for your vector space. Sorry is go further and more in depth but I'm at work and typing this on my phone.

http://en.m.wikipedia.org/wiki/Inner_product_space