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I am studying for a linear algebra test and just can't get a lead on this example question :

given an orthogonal base base in Rn {u1...un) and a set of vectors {y1...yn} where SUM( ∥ yi ∥2) < 1

prove that the set {u1+y1...un+yn} is linearly independent

I tried using the same technique used to prove that an orthogonal set is linearly independent,: showing that a linear combination of the set {u1+y1...un+yn} can only be 0 if all arguments are 0,

a1(u1+y1) + .. + an(un+yn) = 0

by taking the inner product of both sides :

SUM(< a1(u1+y1) + .. + an(un+yn),(uj+yj)> = <0,uj+yj>

but I was not able to draw any meaningful conclusion.

a lead anyone ?

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3 
 Sorry: this isn't a site for homework help. Read the FAQ for suggestions as to where to ask. – Kevin Buzzard Mar 20 at 11:51
Also, the statement isn't true, since it could be that all u_i + y_i are 0, if the vectors u_i are very small. – Joel David Hamkins Mar 20 at 12:28
He must have meant orthonormal, not orthogonal. Hint to the OP: If taking an inner product with $u_j+y_j$ doesn't get you anywhere, try an inner product with something else. You have a very small set of candidates. Since one of the assumptions is an inequality, think of other inequalities that might be brought to bear on the problem, perhaps getting you a contradiction. Think, perhaps, of the expansion of $y_j$ in the given basis. There are lots of things you can try. Don't just give up because the first thing you tried didn't work. – Harald Hanche-Olsen Mar 20 at 12:37
2 
 @Harald: if you give one person a hint with their homework, is there a chance that tomorrow we get 10 people asking about their homework? – Kevin Buzzard Mar 20 at 13:08
3 
 @Kevin: You do have a point there. That is one reason I painted the hint in the broadest possible strokes. If you think even that should not be done, I am certainly willing to discuss it. Start a thread on meta if you wish. – Harald Hanche-Olsen Mar 20 at 14:47
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closed as too localized by Kevin Buzzard, Charles Siegel, Gjergji Zaimi, Joel David Hamkins, Harald Hanche-Olsen Mar 20 at 12:32

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