Suppose that{u1,u2, ,um} are non-zero pairwise orthogonal vectors

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I need some direction. I don't have a clue where to start.
Suppose that{u1,u2,...,um} are non-zero pairwise orthogonal vectors (i.e., uj.ui=0 if i doesnt=j) of a subspace W of dimension n. Show that m<=n.
 

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I assume by dot you mean dot product / inner product? It helps if you are really clear about your notation -- eg is W a vector space, ring, module, etc.

Second, in general, one fruitful way to start proofs like this is to take a simple example which you understand well and look at why your theorem is true or false. So examine, say, R^2 and see why any pairwise orthogonal set must be smaller than the dimension of the subspace it is in. One example set might be the usual basis.

Third, generalize.

Another way to start is to think about what is special about a vector space of size N? You should know that N implies several things -- ie the number of elements in a basis, the largest possible linearly independent set, isomorphism to F^n where F is your field, etc.
 
  • #3
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Thanks that helps alot.
So how would I start my notation to my particular problem and do it.
 
  • #4
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Huh? If you want to say in words what you're having trouble expressing in a mathematical way, I'll help, but you need to put in the work to solve this.

Think about the interaction between linear independence and orthogonality.
 

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