So that part makes sense now.
How do we complete the proof if |H| = 1?
The questions were
For the first one I claimed it was true by using a uniqueness result
and by pairing T(v) with 0. But I can't figure out why this argument does not work for the second one?
Why do you think that "uniqueness result" is true? Add anything orthogonal to x to y and you won't change the inner product of x with it.
(Also the truth of b) is (perhaps surprisingly) dependent on the field your vector space is over.)
Last edited by seanieg89; 6 Apr 2017 at 4:44 PM.
Currently studying:
PhD (Pure Mathematics) at ANU
Is there an intuitive explanation for this?
Let T be a linear map on a finite-dimensional inner product space V
Then T is an isometry iff T is unitary
So the question is obviously easy first year stuff. I'd prove linear independence and then use dim(V) = B to deduce that it's a basis.
However, for the linearly independence step
I just want a validity check because I'm having second doubts. Mostly because no solutions made a remark on this.
I differentiated w.r.t t and then subbed in t=2 to prove c1 = 0. (And then repeated this to show c2 = c3 = 0.) Is this ok?
In an inner product space, the norm is defined in terms of the inner product. This means that any operator that preserves the inner product will preserve the norm.
Less obvious is the fact that in an inner product space, the inner product can be written in terms of the induced norm (*). Consequently anything that preserves the norm will preserve the inner product.
Of course, you don't need to prove (*) in order to answer this particular question, but it kind of hits at the heart of the relationship between inner products and induced norms on a real/complex inner product space and is a good exercise.
Note also that finite dimensionality is not required in any of these arguments.
Currently studying:
PhD (Pure Mathematics) at ANU
The question is an extension on c. How can I explain this answer from first-year calculus?
I don't know much about Graph Theory and Group Theory but are they two different topics? Or Different names but same subjects?
I can't be bothered Googling it
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