Simplicial Complexes and Boundary Maps
The idea behind our definitions is that lots of topological spaces can be "triangularized" in such a way that they look sort of like a bunch of "triangles" glued together.
Matrices for Linear Maps
If you already knew some linear algebra before reading my posts, you might be wondering where the heck all the matrices are. The goal of this post is to connect the theory of linear maps and vector spaces to the theory of matrices and computation.
Linear Maps
Just as we decided to study continuous functions between topological spaces and homomorphisms between groups, much of linear algebra is dedicated to the study of linear maps between vector spaces.
Bases and Dimension of Vector Spaces
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.
Vector Spaces (2) - Direct Sums, Span and Linear Independence
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.
Vector Spaces (1) - Subspaces and Sums
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.
Free Abelian Groups
Essentially, free abelian groups give us a rigorous way of talking about formal linear combinartions of some set of generators.
The First Isomorphism Theorem
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!
Normal Subgroups and Quotient Groups
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?
Cosets and Lagrange's Theorem
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.
Group Homomorphisms
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.
Compactness (2) - The Heine-Borel Theorem
In this post, I will prove a series of seemingly random conjectures about compactness. However, each of them will be required in order to prove the main result of this post: The Heine-Borel Theorem. This theorem completely characterizes compact sets in euclidean space.