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MATH2601 Higher Linear Algebra (1 Viewer)

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions



I have most of the proof covered up but I'm getting confused at the last bit.







All I'm really stuck on is how to prove that if v \neq 0 why must alpha be equal to 0
 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions



I have most of the proof covered up but I'm getting confused at the last bit.







All I'm really stuck on is how to prove that if v \neq 0 why must alpha be equal to 0
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions







So I commenced by stating from Lagrange's theorem that |H| is a factor of |G|. However since |G| is prime, the only possibilities are |H| = |G| or |H| = 1

I'm looking at the |H| = |G| part. I want to deduce from |H| = |G| that H = G, so as H is clearly cyclic so must G. But how do I properly justify that H = G?
 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions







So I commenced by stating from Lagrange's theorem that |H| is a factor of |G|. However since |G| is prime, the only possibilities are |H| = |G| or |H| = 1

I'm looking at the |H| = |G| part. I want to deduce from |H| = |G| that H = G, so as H is clearly cyclic so must G. But how do I properly justify that H = G?
Well H has the same number of elements as G and is a subset of G, so H = G. (If S is a set that has only a finite number of elements, then the only subset of S with the same number of elements as S is S itself.)
 
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leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions

So that part makes sense now.

How do we complete the proof if |H| = 1?
 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions

So that part makes sense now.

How do we complete the proof if |H| = 1?
That can only happen if a = e (the identity). Take a to be any other element in G (there must be at least one other element since G has prime order, which implies |G| is at least 2), and the result will follow (since |H| won't be able to be 1).
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions



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?
 

seanieg89

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Re: MATH2601 Linear Algebra/Group Theory Questions

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.)
 
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InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions



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?
For a), we can do it like this: since T is from V -> V and < u, T(v) > = 0 for all u, v in V, for each v, just take u = T(v), and the result will follow.
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions

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
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions





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?
 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions





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?
Yeah, that's OK for showing linear independence.
 

seanieg89

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Re: MATH2601 Linear Algebra/Group Theory Questions

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
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.
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions







The question is an extension on c. How can I explain this answer from first-year calculus?
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions



 

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