The Perpendicular Subspace of R^n: What is it and How is it Defined?

[Shambles]

Homework Statement

The Attempt at a Solution

The terminology in this question confuses me into what I am actually trying to solve. It seems to me that S-perp would naturally be a subspace of real column vectors based on the fact that we specify that S\neq0. It goes on to mention that S-perp is nonempty which seems obvious in the fact that S is not empty and it asks to show that any scalar multiples of vectors within the subset of S-perp will continue to be elements of S-perp.

So I've reached the thought that either
1) This question is ridiculously simply that intends for me to re-state the obvious or
2) I've missed something completely and it actually requires a long proof. Any insights?

 

[Mark44]

Here's an example that might give you some insight. Consider R², and S = {(x, y) | y = 0}. S^perp = {(x, y) | x = 0}. Each element of S^perp in this example is orthogonal to each element of S, and further, it can be shown that S^perp of this example is a subspace of R².

Your problem is similar. You need to show that your S^perp is a subspace of Rⁿ, which means you need to show the following:

1. S^perp contains the 0 vector.
2. If u and v are in S^perp, then u + v is also in S^perp.
3. If u is in S^perp and c is a scalar, then cu is in S^perp.

 

[Shambles]

Thank you for your assistance. I believe I have constructed a response that is adequate by proving the vectors x,u,v to be elements of S-perp by continually showing that the dot product of the vectors with y (being an element of S) to equal 0 showing orthogonality showing them to be an element of S-perp.

 

[Mark44]

Thank you for your assistance. I believe I have constructed a response that is adequate by proving the vectors x,u,v to be elements of S-perp by continually showing that the dot product of the vectors with y (being an element of S) to equal 0 showing orthogonality showing them to be an element of S-perp.

That's not going to cut it. All you have shown (based on your description) is that the vectors x, u, and v are in S^perp; you haven't shown that S^perp is a subspace of Rⁿ. To do that, you need to show the three things I listed in my previous post.

 

[Landau]

Shambles, do you know what a subspace is, or more precisely, how it is defined? That seems to be the problem.