Bases for vector spaces are similar to bases for topological spaces. The idea is that a basis is a small, easy to understand subset of vectors from which it is possible to extrapolate pretty much everything about the vector space as a whole.
I left off last time with an example of a sum of subspaces with a rather important property. Namely, every vector in the sum had a unique representation as a sum of vectors in the component subspaces.
A vector space is a special kind of set containing elements called vectors, which can be added together and scaled in all the ways one would generally expect.
Essentially, free abelian groups give us a rigorous way of talking about formal linear combinartions of some set of generators.
It would be nice if there was some sort of relationship between cosets of the kernel and the image of a homomorphism. Oh wait... there is!
Let's now revisit the quotient set $G/H$, where $H$ is a subgroup of $G$. What we'd really like to do is turn $G/H$ into a group in a meaningful way. What should the group operation be, though?
It's a bit difficult to explain exactly why cosets are so important without working with them for a while first. But as you'll hopefully start to understand within my next few posts, cosets pop up everywhere and are a necessary tool to get anything done in the world of algebra.
A recurring theme in mathematics is that examining the maps between objects is indispensable to understanding those objects themselves. Of course, that depends on choosing the "correct" type of maps.
Now we can think about a different type of "connectedness." Intuitively, if a space is "connected" you should be able to draw a path between any two points in the space. Otherwise, if there are points in the space that cannot be connected by a path, it is "disconnected."
This is a continuation of Constructing the Rational Numbers (1). Before moving forward with the rest of the construction, I'd like to formally change my notation for rational numbers from that of equivalence classes of ordered pairs of integers to that of fractions.
In this post, I'm going to prove the Intermediate Value Theorem and the One-Dimensional Brouwer Fixed Point Theorem, which are two results that are undeniably and unreasonably useful. In order to prove them, however, we will need to study the notion of connectedness.
We work with number systems every day, but we just sort of take their existence for granted. However, it is possible to construct all of these number systems from scratch.