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Let's have a quick review of 4.2. Quickly, vector spaces.

Vector spaces. So how was that? Well, a vector space is built on a structure that's called a field.

Now that's an algebraic field, not what a physicist calls a field, but what a mathematician calls a field.

So a set K, or let's say a field, an algebraic field, a triple K plus times is a set K,

together with two maps, and two maps plus takes two elements in K and maps it to K,

and multiplication also takes two Ks and maps it to K, that satisfy Cani Cani.

So you saw this before, therefore I'll make it quick. The addition is commutative, it's associative.

There exists a neutral element, let's call it zero in the set, so whatever element in the set it is,

we call it zero, and for every element there exists an inverse.

With respect to addition, the inverse being defined that you take an arbitrary element and you add the inverse,

and then you get the neutral elements. You have to first write down this axiom before you write down that axiom.

Now it's also c times a times n times i times, but this only applies to the set K where you removed the neutral element of addition.

Once you did this, then you have commutativity of multiplication, you have associativity, you have a neutral element,

and the neutral element in that case we call one, if it was named differently we rename it one,

and then the inverse, now you see why we had to take out the zero.

Yes?

Why do we have to explicitly take out the zero, we can't we just choose to ignore it?

Yes, that is the meaning of taking it out.

Okay, I would still leave it in the...

How do you ignore it? You close your eyes when it comes along, well that's the close your eyes operator here.

Okay, I told you.

Okay, no seriously, I mean if you take zero it doesn't have an inverse. So unless you take this out...

Okay, right, for zero itself.

That's the idea. Okay, do I need to spell this out more, is it clear?

Okay, it's the definition of a field, and these are the rules you learned in high school.

Okay, that's a field.

Now, remark, a weaker notion to become, I should add wildly important, to become important later, namely when we consider modules,

is a so-called ring, and the ring comes very much like a field, it is a set equipped with two operations plus and times of the same shape,

but now, but now this only satisfies C, A, N, I for plus, so it stays a commutative or abelian group with respect to plus,

but of these guys here, the commutativity is gone, the associativity needs to stay, the neutral element needs to stay,

well, they're even rings without neutral element, so a unit ring, mathematicians would say, we take them with unit,

and most importantly, the inverses are gone, so you no longer require these, because you no longer require these,

that means of course every field is a ring, but they are rings that are not fields.

Okay, example, first grade high school mathematics, you already know what negative integers are,

you can add them and you can multiply them, that satisfies Cany, it would be commutative, so that would be a commutative ring,

that's a special beast, not special beast, it's a nice ring, it's a commutative ring,

but famously, what is the multiplicative inverse of two, it's one half, map, it's not in there, okay, so this guy is rings,

so you actually earlier in your life learned how to calculate with rings before you learned how to calculate with fields,

okay, so rings are really simple, okay, and the field is very special, and obviously stuff like R plus times,

but also Q and so on, these are all fields, very special rings, okay, but that will become important later.

Another example for a non-commutative ring would be the m cross m matrices over the real numbers,

because you know how to add them and you know how to multiply them by matrix multiplication,

say, you know matrix multiplication is not commutative and you know not every matrix has an inverse,

I mean if the determinant is zero, it doesn't have an inverse, okay, that means the m cross m matrices over R are a ring as defined here, okay,

so it's a very natural structure, you see rings everywhere, function, well, we come to that later, okay, good, so but now we zero in on the fields,

and then we have a definition, a vector, a k vector space, a k vector space where k is a field, that's important,

a k vector space is also a triple, and now I circle the operations because they are not the operations of k,

so the field k has operations plus and times, and this plus is not this plus, and this times is not this times, full stop,

totally different structure, okay, where these stay as they were defined before, where the plus, the shock plus takes two elements in the vector space,