MATH2601 Higher Linear Algebra (1 Viewer)

leehuan

Well-Known Member

Hint was to use the Bezout property but I have no idea to use it. I assume it's related to proving the existence of an inverse because that was the only bit I had trouble proving. (Associativity is just a repeat proof and the identity element is obviously 1)
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Side note - Am not sure why my threads had to be moved here.

Last edited by a moderator:

seanieg89

Well-Known Member
Re: Linear Algebra

Suppose the integer a is a representative of a nonzero equivalence class in Z_p.

Then a is coprime to p and hence am+pn=1 for some integers m and n.

Projecting to equivalence classes we get [a][m]=1 (mod p). ([z] denotes the equivalence class in Z_p of the integer z.)

I.e. every element of U_p has an inverse.

• leehuan

seanieg89

Well-Known Member
Re: Linear Algebra

In fact this argument straightforwardly generalises to tell us that the subset of Z_n consisting of only the equivalence classes coprime to n form a group w.r.t. multiplication.

This yields Euler's theorem just as the version originally posted will yield Fermat's little theorem.

• InteGrand and leehuan

seanieg89

Well-Known Member
Re: Linear Algebra

Ps this isn't really linear algebra at all.

• InteGrand

dan964

MOD
Moderator
Re: Linear Algebra

Ps this isn't really linear algebra at all.
thread title updated to "Group Theory" so no longer misleading • InteGrand

leehuan

Well-Known Member
Re: Group Theory

I am well aware that group theory may not fall under the umbrella of 'linear algebra'. However, in my course (MATH2601), it is the first thing that they choose to teach, hence I set the thread up with that title. I would like my thread name back so I can reserve this thread for all of my math2601 questions. Thanks.

• InteGrand

Green Yoda

Hi Φ
Moderator
Re: Group Theory

I am well aware that group theory may not fall under the umbrella of 'linear algebra'. However, in my course (MATH2601), it is the first thing that they choose to teach, hence I set the thread up with that title. I would like my thread name back so I can reserve this thread for all of my math2601 questions. Thanks.
When you double click on your thread when viewing from "whats new" you can change the name yourself.

• leehuan

leehuan

Well-Known Member
Re: Group Theory

When you double click on your thread when viewing from "whats new" you can change the name yourself.
I know that. I just don't want to appear as though I am undoing a moderator action without permission.

dan964

MOD
Moderator
Re: Group Theory

I know that. I just don't want to appear as though I am undoing a moderator action without permission.
its totally fine to edit the thread yourself I have renamed it again so that it is useful for other users of the site.

• leehuan

leehuan

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

Just out of curiosity, what's an example of a vector space and/or a field which is defined in a way, that does not use the conventional means of addition and (scalar) multiplication?

Drsoccerball

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

Prove:

InteGrand

Well-Known Member
Last edited:
• Drsoccerball

Cult of Personality
Re: MATH2601 Linear Algebra/Group Theory Questions

Just out of curiosity, what's an example of a vector space and/or a field which is defined in a way, that does not use the conventional means of addition and (scalar) multiplication?
there are probably some real valued matrix examples

• leehuan

Drsoccerball

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

Let G be a group with identity e. Prove that if x^2 = e for all x in G then G is abelian.

InteGrand

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

Let G be a group with identity e. Prove that if x^2 = e for all x in G then G is abelian.

• Drsoccerball

Drsoccerball

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

Didn't notice that an element is it's own inverse thanks • InteGrand

leehuan

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

Didn't notice that an element is it's own inverse thanks It's basically this

Having used the associativity axiom and the definition of the identity element.

Right-operating also works

-insert title here-
Re: MATH2601 Linear Algebra/Group Theory Questions

This is strictly for Leehuan's understanding, but here's a concrete example to flesh out for the non-standard addition operation question:

and more generally:

for a bijective function φ(x):

• leehuan

leehuan

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

InteGrand

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

Since V is finite dimensional, by definition V has a finite spanning set S, and this set S also spans W, so W is also finite dimensional.

• Drsoccerball and leehuan