# MATH2601 Higher Linear Algebra (1 Viewer)

#### leehuan

##### Well-Known Member
$\bg_white \text{Define an operation}*\text{over }\mathbb{Z}_n\text{ where }x*y=xy \mod n\\ \text{Proven earlier: This operation is binary, commutative and associative}$

$\bg_white \text{Let }p\in \mathbb{Z}\text{ be prime. Show that }\mathbb{U}_p=\mathbb{Z}_p - \{ 0\}\text{ is a group under the operation }*\text{ defined above.}$

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.

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#### 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.

#### 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.

#### seanieg89

##### Well-Known Member
Re: Linear Algebra

Ps this isn't really linear algebra at all.

#### dan964

##### MOD
Moderator
Re: Linear Algebra

Ps this isn't really linear algebra at all.

#### 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.

#### Green Yoda

##### Hi Φ
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

##### 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

##### 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: $\bg_white (a^{-1})^{-1} = a$

#### InteGrand

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

Prove: $\bg_white (a^{-1})^{-1} = a$
$\bg_white \noindent What's the context? Assuming it's in a group say, note that by definition, aa^{-1} = a^{-1}a = e, where e is the identity element. Hence a is an inverse of a^{-1} (ask yourself what it really means for something to be an inverse of a^{-1} and you'll see from above that a satisfies the requirements). By uniqueness of inverses in a group, a is \emph{the} inverse of a^{-1}, of course.$

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##### 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

#### 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.
$\bg_white \noindent Let a,b \in G. We need to show that ab = ba. We know from the question's assumption that x = x^{-1} for all x\in G. Thus$

\bg_white \begin{align*} ab &= \left(ab\right)^{-1} \\ &= b^{-1}a^{-1} \\ &= ba.\end{align*}

$\bg_white \noindent This completes the proof.$

#### Drsoccerball

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

$\bg_white \noindent Let a,b \in G. We need to show that ab = ba. We know from the question's assumption that x = x^{-1} for all x\in G. Thus$

\bg_white \begin{align*} ab &= \left(ab\right)^{-1} \\ &= b^{-1}a^{-1} \\ &= ba.\end{align*}

$\bg_white \noindent This completes the proof.$
Didn't notice that an element is it's own inverse thanks

#### 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

\bg_white \begin{align*}x^2&=e\\ \implies (x^{-1}x)x&=x^{-1}e\\ \implies ex&=x^{-1}\\ \implies x&=x^{-1}\end{align*}

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:

$\bg_white a \oplus b = \sinh{(\sinh^{-1}{a} + \sinh^{-1}{b})}$

and more generally:

for a bijective function φ(x):

$\bg_white a \oplus b = \varphi(\varphi^{-1}(a)+\varphi^{-1}(b))$

#### leehuan

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

$\bg_white \text{Prove that if }V\text{ is a finite dimensional vector space and }W\text{ is a subspace of }V,\text{ then }W\text{ is also finite dimensional.}$

#### InteGrand

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

$\bg_white \text{Prove that if }V\text{ is a finite dimensional vector space and }W\text{ is a subspace of }V,\text{ then }W\text{ is also finite dimensional.}$
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.