Eric Shapiro

38 posts
Mar 19, 2019

Bases and Dimension of 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.

Mar 11, 2019

Free Abelian Groups

Free Abelian Groups

Essentially, free abelian groups give us a rigorous way of talking about formal linear combinartions of some set of generators.

Mar 5, 2019

Normal Subgroups and Quotient Groups

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?

Mar 4, 2019

Cosets and Lagrange's Theorem

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.

Mar 2, 2019

Group Homomorphisms

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.

Feb 26, 2019

Path Connectedness

Path Connectedness

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."

Feb 21, 2019

Constructing the Rational Numbers (2)

Constructing the Rational Numbers (2)

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.

Feb 21, 2019

Connectedness

Connectedness

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.

Feb 19, 2019

Constructing the Rational Numbers (1)

Constructing the Rational Numbers (1)

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.