Fall 2013
MATH 504: Abstract linear algebra Branko Ćurgus
Monday, December 9, 2013
I made small revisions of the notes that you emailed me yesterday. Here is the resulting
pdf file and here is the corresponding
TeX file. One proof in the notes was wrong and I pointed that out. I did not check all the details. The notes need to be polished further, but it seems to me that they summarize well what we did.
Thursday, December 5, 2013
Today I mentioned that it might be an interesting idea of try to represent a Jordan canonical form of an operator using colors. Below I will present one attempt to do that.
Let $T$ be an operator on an $n$-dimensional complex vector space. Suppose that $T$ has $k$ distinct eigenvalues $\lambda_1, \ldots, \lambda_k$. For each eigenvalue $\lambda_j$ of $T$ there corresponds a unique positive number $m_j$ and a unique nonincreasing $m_j$-tuple $\bigl(q_{j,1},\ldots,q_{j,m_j}\bigr)$ of positive integers such that there exists a basis of the null space of $(T- \lambda_j I)^n$ which consists of $m_j$ Jordan chains (of $T$ corresponding to $\lambda_j$) of lengths $q_{j,1},\ldots,q_{j,m_j}$.
The distinct eigenvalues $\lambda_j$, $j \in \{1,\ldots,k\}$, with the corresponding $m_j$-tuple $\bigl(q_{j,1},\ldots,q_{j,m_j}\bigr)$, determine the Jordan canonical form of the operator $T$.
In the figures below I represent a Jordan canonical form using colored squares instead on numbers. White squares stand for $0$s, black squares stand for $1$s, and the squares of various colors represent distinct eigenvalues.
4 distinct eigenvalues with lengths of Jordan chains
(4,3,2,1,1), (5, 2, 2), (7), (1, 1, 1)
3 distinct eigenvalues with lengths of Jordan chains
(6, 4), (3, 3, 3), (8, 2, 1)
6 distinct eigenvalues with lengths of Jordan chains
(4, 2), (2, 2, 1), (2, 1, 1), (3, 2, 2), (3, 1), (2,
2)
3 distinct eigenvalues with lengths of Jordan chains
(15), (8), (7)
Tuesday, November 26, 2013
Next week we will do the Jordan canonical form of a linear operator.
My notes differ from the presentation in the textbook. This file is needs some polishing. Please email me with errors that you catch.
I made same small corrections to the notes posted before. Here is the resulting
pdf file and here is the corresponding
TeX file.
Tuesday, November 19, 2013
I further revised the notes that you typed up. I also corrected several typos in the first part. I added my proof of the spectral theorem that we did today. Here is the resulting
pdf file and here is the corresponding
TeX file.
Sunday, November 17, 2013
I revised most of the notes that you emailed me. Here is the resulting
pdf file and here is the corresponding
TeX file.
I updated the
Bases file with the content of Chapter 2. I added one more proof of Theorem 2.3. This theorem states that every finite dimensional space has a basis. I also fixed the error in the proof of Steinitz exchange lemma pointed out by Robert.
Monday, October 21, 2013
I am posting a corrected version of
Assignment 1.
Here is the corresponding TeX file.
Sunday, October 20, 2013
Here is a pdf with my notes about linear operators. This is a draft with many mistakes; please report them to me.
Monday, September 30, 2013
Here is a pdf file with the content of Chapter 2 with formal proofs of statements from the book that I consider not to be proved formally in the book. Read both the book and my notes and find the corresponding statements. On exams I expect you to be able to present formal proofs.
Here is the LaTeX file that I used to produce Bases2013.pdf. Right-click on the underlined word "Here"; in the pop-up menu that appears, your browser will offer you to save the file in your directory. Make sure that you save it with the exactly same name, Bases2013.TeX.
I encourage you to learn LaTeX. LaTeX is a document preparation system which is widely used by mathematicians. In fact it is the standard for professional mathematical publications. It is completely free. A good starting point would be the LaTeX class on Thursday, October 3, from 4:00 to 5:30 in BH 215. The instructors are Andrew Wray and Robert Brokken.
There is an abundance of information about LaTeX on the web. Here are a few sites to start with.
If you have a problem with any aspect of LaTeX ask me or any LaTeX enthusiast. LaTeX community is very willing to help you to discover this amazing tool. Also, keep on googleing and you will find a site that will answer your question.