Proving Orthogonality of a Vector to a Spanning Set of a Subspace

[kehler]

Homework Statement

Prove that a vector k is in 'W perp' if and only if k is orthogonal to every vector in the spanning set of W, where W is a subspace of Rⁿ

The Attempt at a Solution

It's so obviously true that I don't know how to prove it! :S

Here's what I did:
Let {w₁, w<sub>2SUB], ... wm} be a spanning for W.
Let w be a vector in W where
w = c1w1 + c2w2 + ... + cmwm
and all the weights c1, c2,...,cm are nonzero
Suppose k is in 'W perp' but is not orthogonal to every vector in the spanning set.
Then k.w
= k. (c1w1 + c2w2 + ... + cmwm)
= c1k.w1 + c2k.w2 +... + cmk.wm
= a, where a is a non-zero number
But to be in 'W perp', the dot product of k and any vector in W must be zero
Therefore, k has to be orthogonal to every vector in the spanning set of W to be in 'W perp"

Is this right?? I can't think of any other way to do it :S</sub>

 

[HallsofIvy]

Your definition of 'Wperp' is the space of all vector orthogonal to every member of W, right? And you are given that a vector is orthogonal to every member of a spanning set?

Do a direct solution, not indirect. If k.wi= 0, what is k.w?

 

[kehler]

Yep 'W perp' (W perpendicular) is the set of all vectors orthogonal to W.
Oops I realized I made some typo errors. The 'x's in the last two lines were meant to be 'k's. I've edited it now.
What do you mean by a direct solution?
And what's 'wi'? :S

 

[gabbagabbahey]

He means that your basis set is \{ \vec{w}_i \}= \{\vec{w}_1,\vec{w}_2, \ldots \vec{w}_n \} and if \vec{k} is perpendicular to each of your spanning vectors, \vec{k} \cdot \vec{w}_i=0 \quad \forall i \in [1,n]. What does that make \vec{k} \cdot \vec{w}?

And if \vec{k} is perpendicular to every vector in W, then \vec{k} \cdot \vec{w}=0. What does that make \vec{k} \cdot \vec{w}_i=0 for each i?

 

[kehler]

If k is perpendicular to each of the spanning vectors, then k.w = 0
and if k is perpendicular to every vector in W, then k.wi = 0

Is that all I have to write? :S

 

[gabbagabbahey]

Well, you should explicitly show that each of those claims is true; but other than that, yes; since any vector for which k.w is zero, is in W_perp.

 

[kehler]

Oh ok cool :). Thanks guys.