Is the Set of Orthogonal Vectors to Any Non-Zero Vector a Subspace?

[ercagpince]

Homework Statement

In a space $$V^{n}$$ , prove that the set of all vectors
$$\left\{|V^{1}_{\bot}> |V^{2}_{\bot}> |V^{3}_{\bot}> ... \right\}$$
orthogonal to any $$|V> \neq 0$$ , form a subspace $$V^{n-1}$$

Homework Equations

The Attempt at a Solution

I tried to make a linear combination from that set and product with <V|, I yielded nothing logical , at least I didn't understand the outcome .
I wrote <V| as linear combination of basis in V^n vector space , I thought
that since the |V> and those vectors share the same vector space , it might be possible that they have the same orthogonal basis (just an assumption which is probably false) .

All it left to me the product of components of these vectors as a matrix , but as i said before I have no clue that I am doing the right thing to solve this problem .

 

[Dick]

X is orthogonal to V if <V|X>=0. To show such vectors form a subspace you just have to show if X and Y are orthogonal to V and c is a scalar then cA and A+B are also orthogonal to V.

 

[ercagpince]

What are A and B ?

 

[Dick]

What are A and B ?

Ooops. I meant show cX and X+Y are orthogonal to V. Forgot my notation.

 

[ercagpince]

how can I show it ?
That is the problem actually .

 

[Dick]

Use properties of the inner product! <V|(X+Y)>=<V|X>+<V|Y>, for example.

 

[HallsofIvy]

Why in the world is this under "physics"? This is a pretty standard Linear Algebra question!

 

[ercagpince]

I saw this problem on a quantum mechanics textbook , that's why I subscribed it in here .

Thank you dick by the way .