Maths Uni Chat

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A vector is some element of a set (V) that has two functions related to it (+ and .) and + : V^2 -> V and . KxV -> V and K is some structure similar to the real numbers (actually much weaker than that, I have no clue how much group theory they do there but assuming that K is the set of real numbers usually suffices in first year). + and . satisfy some shizzle. You probs know what it is (vector space axioms).

If you take K to be R and V to be the set of all functiondsfrom R to R then the quadruple (V,R,+,.) is a vector space and we class elements from R to R as vectors.

I think this answered your problem though ask once more if it has not and hopefully I can help.

Reply 4801

MrShifty

Simplicity

I can understand how vectors can be just a list like (a,b,c,...,z). But, how can it be a function.

You're thinking about it the wrong way round. Instead of thinking of how a vector can be a function, it's better to think of how certain functions can form a vector space (and thus be considered 'vectors').

For instance, linear maps between from a vector space to its scalar field obviously form a vector space (the dual space) because they form an abelian group and a scalar multiplication is defined satisfying the axioms.

Reply 4802

Hedgeman49

BJack

Trains in a fish tank? This won't end well. :erm:

It's abstract... it's whatever the **** I want it to be. :p:

Reply 4803

MrShifty

Simplicity

As I'm unclear what a vector is. Its like a point here with an arrow pointing towards it.

That's an example of a vector (or, more specifically, an example of how a vector in a particular vector space is envisaged).

So its something with magnitude and direction,

Not really. 'Magnitude' and 'direction' only really make sense if they're defined. For instance, such concepts make sense for vector spaces like R, or R^n, as we have a sense of what magnitude and direction is in those cases (although, strictly speaking, they're not concepts that are intrinsic to the notion of a vector space or vectors).

What you're asking doesn't actually make much sense. Think for a moment about this analogous question: "What does an element of a group look like?" or, even more basic, "What does an element of a set look like?". The obvious answer is that it depends entirely on what group and what set we're talking about, and we should be prepared to accept that even if we restrict our attention to a particular set or group, the element in question might not 'look' like anything in the intuitive or naive sense.

This is exactly the same as the question you're asking, and suffers the same problems; a vector is simply an element of a vector space. For some vector spaces, a vector will be a function, for others a point in the real plane (in which case we can easily imagine what it looks like).

Reply 4804

around

Simplicity

What does a vector space look like?

A big box filled with arrows.

Reply 4805

Dadeyemi

how long is a piece of string?

Reply 4806

My Alt

Edenr

He's not your guy, mate.

I'm not your buddy, guy.

Reply 4807

My Alt

Dadeyemi

how long is a piece of string?

Your Mum.

Reply 4808

My Alt

around

A big box filled with arrows.

Pretty much exactly how I imagine it :biggrin:

Reply 4809

v-zero

BJack

Trains in a fish tank? This won't end well.

What, fish don't deserve a quality transport network? That's fishism...

Reply 4810

Simplicity

MrShifty

You're thinking about it the wrong way round. Instead of thinking of how a vector can be a function, it's better to think of how certain functions can form a vector space (and thus be considered 'vectors').

For instance, linear maps between from a vector space to its scalar field obviously form a vector space (the dual space) because they form an abelian group and a scalar multiplication is defined satisfying the axioms.

Yeah, that cleared things up. Its confusing according to wikipedia the vectors we learn in A levels are actually called Euclidean Vectors, so that was probably confusing me. Surely, all vector spaces are isomorphic to some vector space of Euclidean vectors?

Also, is there something called a non euclidean vector and does that form a vector space?

According to wikipedia the addition is the abelain group part. Is a vector space a ring?

P.S. I'm probably never going to use linear algebra again. Looking at later modules of uni just now. But, meah.

P.P.S. How does everyone manage there time studying? As like 90% of my study time is to algebra. Thats why I suck at everything that is non algebraic, even then I'm not even that good at algebra. Its just its so easier to read like a book on abstract algebra as it makes sense, then I try to read a book on say PDEs or analysis and its like arrrghh.

Reply 4811

MrShifty

Simplicity

Surely, all vector spaces are isomorphic to some vector space of Euclidean vectors?

Nope. Every vector space over the real numbers is isomorphic to a vector space of Euclidean vectors, because a vector space over the reals of dimension n is isomorphic to R^n (because two vector spaces are isomorphic iff they have the same dimension). However, not every vector space is isomorphic to such a vector space. Remember, the definition of an isomorphism necessitates that both vector spaces are defined over the same field.

If a vector space is defined over a finite field, then it's clearly not isomorphic to R^d for any d (apart from anything else, you can't even define a linear map between the two!).

According to wikipedia the addition is the abelain group part. Is a vector space a ring?

Not necessarily. A ring has multiplication defined for it's elements, whereas a vector space need only have 'scalar multiplication' defined (that is multiplication of its elements by elements from the field over which it's defined).

A field (that is a ring with all the added stucture of a field) can be a vector space defined over itself (i.e. R with the usual multiplication of fields).

P.S. I'm probably never going to use linear algebra again. Looking at later modules of uni just now. But, meah.

I wouldn't bet on it! Linear algebra has a habit of creeping into everything.

Reply 4812

assmaster

v-zero

What, fish don't deserve a quality transport network? That's fishism...

Australian Nazi experiences?

Reply 4813

Simplicity

MrShifty

Nope. Every vector space over the real numbers is isomorphic to a vector space of Euclidean vectors, because a vector space over the reals of dimension n is isomorphic to R^n (because two vector spaces are isomorphic iff they have the same dimension). However, not every vector space is isomorphic to such a vector space. Remember, the definition of an isomorphism necessitates that both vector spaces are defined over the same field.

If a vector space is defined over a finite field, then it's clearly not isomorphic to R^d for any d (apart from anything else, you can't even define a linear map between the two!).

Not necessarily. A ring has multiplication defined for it's elements, whereas a vector space need only have 'scalar multiplication' defined (that is multiplication of its elements by elements from the field over which it's defined).

A field (that is a ring with all the added stucture of a field) can be a vector space defined over itself (i.e. R with the usual multiplication of fields).

I wouldn't bet on it! Linear algebra has a habit of creeping into everything.

Huh, hmm I didn't know that. I thought it was the same definition of ismorphism like in group theory. But, instead of the homorphic priniciple instead you have linear map.

You can't? Surely, there is a linear map between vector space of field R and a vector space of field C.

A field defined over itself?(R with the field R is a vector space?)

I remeber you saying you forgot all of linear algebra. Something like that.

My Alt

Pokermon advice

So I'm playing heartgold. Whats the best steal pokermon? I got a Onix but I was wondering how do you turn the Onix into a Steelix. Something about trading it, but surely I would lose the Onix?

Also, what do you think of Ampharos?

Reply 4814

v-zero

assmaster

Australian Nazi experiences?

Je ne comprends pas.

Reply 4815

MrShifty

Simplicity

You can't? Surely, there is a linear map between vector space of field R and a vector space of field C.

Not really. It's not a bad question though! Strictly speaking, linear maps are only defined for vector spaces over the same field, that the basic definition (the proper name of these maps are "F-linear", where F is the field of definition for the two vector spaces.

However, as your example suggests there are occasions when you can get away with defining a linear map between vector spaces over different fields. Unfortunately, that particular example isn't very interesting!

Say V is an R vector space and W a C vector space. The condition that f(ax)=a(fx) in the definition of a linear map means that whilst technically we're mapping an R-space into a C-space, all we're really doing is treating W as a vector space over R (which we can do, because R is a subfield of C) and mapping into that - so after all that we're only really constructing an R-Linear map between two vector spaces over R

A field defined over itself?(R with the field R is a vector space?)

Yeah. As in take the elements of the vector space to be the elements of R, and define scalar multiplication R X R -> R to be ordinary multiplication within R. What you've got is a vector space.

All a vector space over a field F is an abelian group V with a scalar multiplication F X V -> V defined such that the distributive, compatibility and multiplicative unity axioms are satisfied (i.e. all the extra 'junk' in the definition of a vector space).

R is certainly an abelian (additive) group, and the rest of the axioms are immediately satisfied by the properties of a field - so it's a (one dimensional) vector space over itself.

I remeber you saying you forgot all of linear algebra. Something like that.

I did forget a fair bit, but certainly not all - partly because it kept popping up all the time!

Reply 4816

My Alt

Simplicity

Best Steel Pokemon? Scizor without doubt, at least competitively. Metagross, Lucario and Heatran are all pretty beast. But Scizor is the only one oyu can get in HG without trading iirc.

To get Steelix trade Onix to another game whilst holding metal coat. Then trade it back to you once it's evolved. So use your own game or trade with someone you can trust.

Ampharos is k, pretty useful if you're trying to finish the game. I used ampharos in HG when I played it I think.