# Fall 2011 MATH 504: Abstract linear algebraBranko Ćurgus

Monday, December 5, 2011

Wednesday, November 30, 2011

• Again, updated notes on the Jordan normal form.
• The problems in Chapter 8 which are relevant to what we covered are the following: 1, 2, 3, 5--16,20,31.

Monday, November 21, 2011

• Updated notes on the Jordan normal form.

Friday, November 17, 2011

Thursday, November 17, 2011

• Here is the fourth problem from the exam: Let $({\mathcal V}, \langle\,\cdot,\cdot\, \rangle)$ be a finite dimensional inner product space over $\mathbb C$ and let $T \in {\mathcal L}({\mathcal V})$.
(a) Prove: If ${\mathcal B} = \{u_1, \ldots, u_n\}$ is an orthonormal basis of ${\mathcal V}$, then $\sum_{k=1}^n \, \|T^* u_k\|^2 = \sum_{k=1}^n \, \|T u_k\|^2.$ (b) Prove: There exists $v\in {\mathcal V}$ such that $\|T^* v\| < \|T v\|$ if and only if there exists $w\in {\mathcal V}$ such that $\|T^* w\| > \|T w\|$.

Tuesday, November 15, 2011

Thursday, October 27, 2011

Wednesday, October 5, 2011

• I just updated Bases file. And this is the corresponding LaTeX file.

Thursday, September 29, 2011

• I keep making small modifications in the files that I posted on Tuesday, September 27. The pdf file is stamped with the date and time. Although, the modifications are small, check occasionally that you have the latest version.
• Here is a text file with the instructions how to install LaTeX on your computer.

Tuesday, September 27, 2011

• 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. I expect that the file that I am posting has errors. If you find an error please report it to me. I will give extra credit for each report. The extra credit will vary with the nature of the reported error.
• This is the LaTeX file that I used to produce Bases2011.pdf. Right-click on the underlined word "This"; 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, Bases2011.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 Wednesday the 28th of September from 4:00 to 5:30 in CF162/164. The instructor is Brian Schiller.
• 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.

Friday, September 23, 2011