# Math 15a: Applied Linear Algebra

## Spring 2004

Instructor: Prof. Daniel Ruberman
E-mail: ruberman@brandeis.edu
Office: Goldsmith 310
Phone: 736-3074
Class schedule: Mon Wed Thu 10-11, Goldsmith 317.
Office hours: Monday 11-12, Wednesday 1-2, Thursday 11-12, and by appointment. My schedule for spring 2004.
Textbook: Linear Algebra with applications (2nd edition) by Otto Bretscher.

## Syllabus

• Chapter 1: Linear systems, matrices, Gaussian elimination.
• Chapter 2: Linear transformations, matrices, inverses. Geometrical interpretations.
• Chapter 3: Vectors, subspaces of Rn, dimension, kernel and image of a linear transformation.
• Chapter 4: Orthogonality, Gram-Schmidt process, orthogonal matrices, least squares.
• Chapter 5: Determinants, Cramer's rule.
• Chapter 6: Eigenvalues, eigenvectors.
• Other topics (time permitting): Coordinate systems, diagonalization of matrices. Applications to linear differential equations.

• There will be 2 midterm exams, 2 quizzes, and a 3-hour final exam. The schedule for the in-class exams and quizzes will be posted shortly; the final is Wednesday, May 12, 9:15 am - 12:15 pm (Block C).
• Homework will be assigned and collected weekly and will be graded and returned to you. You may discuss homework with others, or with me, but you must write up your solutions by yourself.
Homework 10%
Quizzes 15%
Mid-term exams 40%
Final exam 35%

• Homework assignments will be available from this page. (Note: The homework assignments and review sheets have not yet been posted. Please check back at the beginning of the semester.) Some of this will be in pdf format, for which you will need the free Acrobat Reader.
• Here are a couple of handouts that I distributed in class. In Chapter 5.4 we studied least-squares and curve-fitting. In chapter 7, we did an example of a predator-prey model (cats and mice). Here is the graph that I handed out in class

## Homework assignments.

In addition to the even-numbered homework, which are to be handed in for grading, I have suggested some odd-numbered problems for you to do for practice. The solutions to these are in the back of the book, so they won't be graded; you can write the solutions on the same paper as the even ones if you find that to be convenient. Sometimes (eg the first assignment) there will be some problems that I made up; you can find them by clicking the green button. Solutions will be posted after the problems are turned in, and are accessed by clicking the orange buttons.

No. Date due Sections to read Problems Solutions
1
Jan. 15   1.1

2   Jan. 22   1.1
1.2
1.1: # 6, (7), 20, (21), (27), 36.
1.2: # 2, (9), 12, 18, (23), 24, 34.

3
Jan. 29
Appendix A, 441-446
1.3
2.1
1.3: # 2, (3), 4, 6, 10, (17), 18, (19), 22, (27), 28, (47), 48.
2.1: # (1), 2, (3), 4, 6 (more to come on next HW).
4
Feb. 5
2.1
2.2
2.1: # 8, 16, (19), (35), 38.
2.2: # (1), 2, 6, (7), 10, 12, (13), (19), 20, (31), 32, 38, 40, 42.
5
Feb. 12
2.3
2.4
2.1: # (13), 14.
2.3: # 4, (5), 6, (7), 18, (29), 34, (35), 40, (41)
2.4: # (3), 4, (7), 16, 18, 24, 28, (37), 40, 44, (45)
6
Weds.
Feb. 25
2.4
3.1
2.4: # (37), 38, (45), 46.
3.1: # 4, (5), 12, (15), 16, (23), 24, 32, (33), (39), 40.
7
March 4
3.2
3.3
3.2: 2, (3), 4, 6, 14, (17), 18, (19), (23), 34, 36, 32
3.3: (3 & 13), 6&16 , (10 & 19), 8& 20. For each of these, check consistency with fact 3.3.9 (rank + nullity theorem). 32, (35), 38, (39).
8
March 11
3.4 (through p. 138)
5.1
3.4: (1), 2, 4, (5), (11), 12, 22, (24). This last problem is really nice, but is optional.
3.3: 42 (if you didn't do it on #7) 5.1: 2, (5), 10, 12, (13), 14, 16, (17), (23), 26, (27), (37).
9
March 18
5.2 (through p. 197)
5.3
5.2:(3), 6, 14 (ignore the QR stuff) (29), 32, (33), 34, (35).
5.3: (1), 2, (3), 4, 6, (7), 10, (13), (15), 16, 20.
10
Weds.
March 24
5.4
Least squares handout
5.4: (1), 2, (19), 20, 30, (31), 32, (33).
11
April 1
6.1
6.2
6.1: 4, (9), 12, 22, (23), (27), 28, (39).
6.2: 2, (5), 8, (9-13), (15), 16, (23), 24, (25), 26, 39.
12
Fri. April 16 (Brandeis Monday)
6.3 (Volume) 6.3: (1), 2, (9), (11), (13), 14, 16, 18.
13
April 22
7.1
7.2
7.1: (1), 2, 4, 8, (9), (15), 18, (19), (33), 36
7.2: (1), 2, 4, 10, (15), 20, (21), 22, (25), 26.
14
Mon. May 3
7.3
7.4 (through Example 4)
7.3: (5), 6,8, 14, 18, (19), 22, 26, (33), 34, (41), 44.
7.4: 2, (3), (11), 12, (15), 16.

## Exam schedule and solution sheets

Event Date Review
sheet
Review
solutions
Text sections
Quiz 1
Thursday Feb. 5
N/A
1.1-2.1
Quiz 2
Thursday, April 22
N/A
5.4 - 6.3
Exam 1
Thursday, Feb 26
7:30-9pm
Chapters 1
and 2
Exam 2
Thursday, March 25
7:30-9pm
Chapters 3
and 5 (not 5.4)
Final  May 12, 9:15 am - 12:15 pm
Room 317 Goldsmith
Chapters 1-3, (skip 4), 5 (except for 5.5), 6 (no Cramer's rule in 6.3), and 7 (in 7.4, only up through p. 334).