I have a really long homework question that I need help with. There are a couple of parts to it so I appreciate anyone that can help out.

It starts by just stating that a basis is an orthonormal basis if the following holds for all :

Assuming this, I have to:

a) Show that for any we have:

b) Show that:

is an orthonormal basis in [tex]\mathbb{R}^3[\tex] and expand the vector in the basis .

c) Show that for any :

and for any :

As I said, this seems like a lot to go through and appreciate any help since I literally am stuck!