MATHEMATICS 5101: Linear Mathematics in Finite Dimensions

Information for the section at 9:10 of AU 2017,  Enarson Classroom Bldg 358


Instructor: Dr. Rodica D. Costin
Office: 436 Math Tower
Office hours: MW 10:20-11:15 AM or by appointment
e-mail: costin.10

General information


I will post here lecture notes, as well as homework assignments and other announcements.

Wed Aug 23 We went over Sec.1.1 and 1.2 in the lecture notes 1.Vector Spaces.
Fri Aug 25
We continued from the notes above: Sec. 1.3
Mon Aug 28
You are responsible for reviewing complex numbers (operations, polar form, geometric interpretation, the n'th roots of 1)
                      Exercise: (you do not need to turn it in, but we will use these facts)
                      1. Show that any n'th root of 1 (that is, a number z so that z^n=1) has the form exp(2 pi k i/n) for k integer.
                      2. Denote a=exp(2 pi i/n). Show that 1, a, a^2,..., a^(n-1) are all the roots of 1 (that is, there are n of them, and no more).
                      3. Show that the number a above satisfies 1+a+a^2+...+a^(n-1)=0
                      4. Plot, on separate planes, the roots of 1 for n=2. Then for n=3, then n=4. Then n=5. What do you see?


Wed Aug 30  We started Sec. 2.2.
                      HW 1 due Wed Sept 6 

Fri Sept 1st 
We finished the chapter on Vector Spaces.
Wed Sept 6 
We start talking about determinants. Here are the lecture notes: 2. Determinants (new file)
Here is a Maple example on how to solve systems and produce latex output.
                    HW 2
due Wed Sept 13.
Fri Sept We
continued determinants (Sec. 2.4 and 2.5) with a brief break for a fire alarm.

Mon Sept 11  We finished determinants and started 3. Linear Transformations.
Wed Sept 13  We
did Sec. 2.2 and 2.3 (of Linear Transformations).
Here is HW3
due Wed Sept 20.
Fri Sept 14  Null
space and range. Here is the edited chapter on linear transformations, with changes in Sec. 3.5.
Mon Sept 18  We
did Sec. 3.6, 3.7.

Wed Sept 20  We did Sec. 3.8, 3.9.1. Here is HW4
Fri Sept 22  
We continued...
Mon Sept 25  
and continued... with Sec. 3.11
Wed Sept 27  
Here is a handout  and HW 5 .
We will soon start the chapter on  4.Eigenvalues and Eigenvectors.

Fri Sept 29  
We finished Ch. 3 and started Ch.4, on Eigenvalues and Eigenvectors, Sec. 4.3
Mon Oct 2 
We continued with Sec. 4.2, 4.4...up to 4.12.
Wed Oct 4 
We continued with 4.1.13, 4.1.14 (but did not go over Remark 1. points 3. and 4.)  HW6   and its corresponding handout.
                    Here is the chapter on Eigenvalues and Eigenvectors with revised proof of Theorem 6 and of sections 4.13, 4.14 (I expect more revisions along the way)

Fri Oct 6
Sec 14.14, 15
Mon Oct 9 
We proved that similar matrices have the same characteristic polynomial and we did a review.

MIDTERM: Wednesday Oct. 11. You are allowed to bring a cheat sheet (regular size paper with your own notes, written on both sides, if you wish). Required material: everything!

Topics not to be missed when you review: examples of vector spaces that we often encountered, linear transformations (definition, null space, range, theorems and other facts, inverse, right and left inverse, important examples). Eigenvectors and eigenvalues (definition, calculation, theorems and other facts, application to differential equations).

Wednesday Oct. 11 Midterm!

Have a Nice Break!

Mon Oct 16  We showed that det(AB)=det A det B, here are the notes. And 4.14. 1...3.
Wed Oct 18  Here is HW7. We did 14.6.: 3, 4, 14.7, 5.1, 5.2
Fri Oct 20  We covered 5.3
Mon Oct 23  We covered 5.4, 5.6
Wed Oct 25  HW8  We covered 5.6.1 and 2, also 5.8
Fri Oct 27  plan:  5.9, 5.10-12, 6.2.7, 6.3 then Inner Product Spaces
Mon Oct 30  we continued...
Wed Nov 1
and continued up to page 10. Here is HW 9.
Fri Nov 3 and continued

Mon Nov 6 and continued  up to Sec. 8.1

Wed Nov 8 we continued with 8.2,8.3 and 11.1. Here is HW 10.
             Selfadjoint transformations

Mon Nov 13 and continued with Sec. 11.2-3, 11.5-6

Wed Nov 15 Sec 11.7, 11.9-11, 12.1.1. And here is next HW, due Wed Nov 29,  HW 11
Fri Nov 17 We continued up to 12.3.
Mon Nov 20
We continued up to 12.7
Happy Thanksgiving!
Mon Nov 27 continued with 12.8, 13.1 and part of .2
Wed Nov 29 13.3, 13.4, 14.1. Here is HW 12 it will not be graded but it will help you prepare for the final exam
Fri Dec 1    SVD
, pseudoinverse
See some examples of image compression
Mon Dec 4
Finish Sec. 14, 15.
Wed Dec 6 Review.
Here are some Review Problems  and some Review Questions (we will solve some in class). This will be attached to the exam.
Special Office Hour: Sunday Dec 10, 2-3 PM.

Final exam: Monday Dec 11 10:00am-11:45am according to the university schedule (please check!)
You are allowed to bring a cheat sheet (regular size paper with your own notes, written on both sides, if you wish). Required material: everything!

Other references:

Gilbert Strang's lectures