If is a subspace, you absolutely must have Are you sure you meant
Also, did you mean that is an orthogonal basis for
the task is following. Let's assume that is orthogonal vector set of subspace and . Also . It's defined that:
So, the problem is to show that is orthogonal to all vectors of .
When the sum is opened, we get:
It seems like formula from Gram-Schmidt process, so in my opinion it must be , because G-S process orthogonalizes vectors. I think I need a little bit more evidence to show the proposition, but I don't know how to continue. Any help is welcome. Thanks!
Well, you'll get the scalar 0, since the result of any dot product is a scalar, right? But yes, you should get zero if they are orthogonal. You can't compute the dot products directly, but you can use various theorems you know about dot products, such as:
for all scalars and vectors and
for all vectors
You also know that is an orthogonal set (all the vectors in it are mutually orthogonal). That tells you something about, say, right?
Just dive in and see what happens.