Category Archives: Linear Algebra

Hilbert Spaces – Part 2

Posted on 05/28/2012 by Ryan

Recall that in the finite dimensional world, two vector spaces are isometrically isomorphic if and only if they have the same dimension. In the last post, I mentioned that, by using the appropriate definition of a basis and dimension, one … Continue reading →

Posted in Analysis, Linear Algebra | 1 Comment

Digression: Hilbert Spaces – Part 1

Posted on 03/15/2012 by Ryan

Hello all, I hope your semesters are going well! I am on spring break this week, so I’ve got a bit of time, and I figured instead of getting a head start on my homework for next week, I’d write … Continue reading →

Posted in Analysis, Linear Algebra | 1 Comment

The Riesz Representation Theorem – Part 2

Posted on 02/28/2012 by Ryan

So I should explain: this is one of those weird weeks where I have actually finished all my homework for the week two days early, so I actually have some time to write on the blog! So, let’s talk about … Continue reading →

Posted in Analysis, Linear Algebra | 3 Comments

The Riesz Representation Theorem – Part 1

Posted on 02/27/2012 by Ryan

Hello everyone! I realize it’s been far too long since I’ve last posted, and I decided I don’t really want to write about Radon-Nikodym anymore. Maybe someday if I get requests I’ll write a couple more posts in that series, … Continue reading →

Posted in Analysis, Linear Algebra | 3 Comments

Determinants – Part 4

Posted on 11/06/2011 by Ryan

Ah we’ve finally arrived! We have defined, played with, and even proved the existence of alternating n-linear forms on a vector space . Even better we have defined the determinant of a linear transformation, a concept not usually seen in … Continue reading →

Posted in Analysis, Linear Algebra | 1 Comment

Basic AC, part 2: Choice in Algebra

Posted on 11/05/2011 by Z Norwood

The outline I (apparently) wrote on the previous post in this series says this post should talk about the Axiom of Choice in algebra, particularly how it affects vector spaces and groups. Before I talk about vector spaces and groups, … Continue reading →

Posted in AC, Linear Algebra, Logic | 11 Comments

Determinants – Part 3

Posted on 10/30/2011 by Ryan

Hello everyone, I hope you are all surviving your semester, wherever you may be! In this post, I want to show that (non-zero) alternating n-linear forms always exist over any vector space of dimension n. We already know that, if … Continue reading →

Posted in Linear Algebra | 1 Comment

Determinants – Part 2

Posted on 10/23/2011 by Ryan

Last post I defined and talked a little bit about alternating n-linear forms on a vector space (or, more precisely, on the n-fold product of with itself). It’s not hard to see that the space of all n-linear forms on … Continue reading →

Posted in Linear Algebra | 9 Comments

Determinants – Part 1

Posted on 10/07/2011 by Ryan

Hello everyone! Welcome to, and thank you for reading, my first non-test post, i.e. my first post about some actual mathematics! Perhaps because my teaching assignment this semester is a homework help session for introductory linear algebra courses (and perhaps … Continue reading →

Posted in Linear Algebra | 3 Comments