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I am having trouble proving the following...
Let B be a basis for a finite dimensional inner product space V.
Prove that <x,z>=0 implies x=0 (the zero vector) for all z in B.
Also prove that <x,z>=<y,z> implies x=y for all z in B.
So far all I have is proof that at least one element of z must be nonzero, hence one element of x is zero, cant get any farther though.
Any help would be appreciated.
Thanks
I think you need to learn to use quantifiers (read and write them) since what you wrote makes little sense as it is.
Anyway from what I get you're trying to prove that iffor all
then
. If it's so then
but a norm is zero iff
, for the second one use the first one with
for all
I actually quantified z to be a vector in B, the basis for V
what I did forget (thanks for pointing it out) is that x and y are vectors in V
But the order is important, your quantifier should go before the implication (check it!), unless of course I understood something else in which case let me know.
Anyway my proof assumes z was in V but it still works since if <,z> is zero on a basis of V then <,w>=0 for all w in V by bilinearity