# Linear Spaces and Matrix Theory. Math 419, Winter 2013

Instructor: Roman Vershynin
Office: 3064 East Hall
E-mail: romanv "at" umich "dot" edu

Class meets: Section 1: Tu, Th 10:10 - 11:30 am, 104 EWRE; Section 2: Tu, Th 12:10 - 1:30 pm, 1024 FXB.

Office hours: Mo, Th 2:30 - 3:40 in 3064 East Hall.

Course description. This course is an introduction to linear algebra and the theory of matrices and linear spaces, with a view toward aplications in science and engineering. Concepts, calculations, and the ability to apply principles to solve problems are emphasized over proofs, but arguments are rigorous. This course should be very useful for students in applied mathematics, and in all areas of science and engineering. Ctools site

Prerequisites: Four mathematics courses beyond Math 110.

Alternatives: Math 417 (less rigorous, more applied); Math 217 (more proof-oriented), Math 513 (even more abstract and sofisticated).

Textbook: Otto Bretscher, Linear algebra with applications. Pearson Prentice Hall, 4th ed., 2009. ISBN: 9780136009269.

• Homework (30%). Late homework will not be accepted. One lowest homework will be dropped.
• Midterm Exam (30%). Tuesday, March 12.
• Final Exam (40%).
Homework will be assigned every Tuesday (scroll down to see it), and it will be collected next Tuesday before class. It is important to do the homework to master the material. Collaboration on homework is allowed, but you must write down your solutions individually. Exams are closed book, and no calculators are allowed.

The following cut-offs guarantee you the course grade: 90%: A, 80%: B, 70%: C, 60%: D.

Course Schedule:

Thursday, January 10
1.1-1.2. Linear systems. Gaussian elimination. Row echelon reduced form.
Exercises (not to be turned in): 1.1 problems 1-28 except 22.
Tuesday, January 15
1.3. Number of solutions. Rank. Matrix algebra.
Exercises (not to be turned in): 1.2 problems 1-29, 34, 36, 39; 1.3 problems 1-48.
Homework 1 (due Tuesday, January 22). Solutions.
Thursday, January 17
2.1, 2.2. Linear transformations. Rotations by an arbitrary angle, projections onto an arbitrary line in R^2.
Tuesday, January 22
2.2 (contd.), 2.3. Reflections about an arbitrary line in R^2. Matrix multiplication.
Exercises (not to be turned in): 2.1 problems 1-7, 16-34, 36-38; 2.2 1-18, 26, 30, 31; 2.3 1-16, 43-48.
Homework 2 (due Tuesday, January 29). Solutions.
Thursday, January 24
2.4. The inverse of a linear transformation. (Determinant was not covered at this time.)
Tuesday, January 29
3.1. Image and kernel.
Exercises (not to be turned in): 2.4 problems 1-20, 28-31, 34-44, 67-75; all Chapter Two Exercises (pp.98-100).
Homework 3 (due Tuesday, February 5). Solutions.
Thursday, January 31
3.1 (contd.) Characterization of invertible matrices. 3.2 (begins). Linear subspaces.
Tuesday, February 5
3.2. Subspaces. Linear independence. Bases.
Exercises (not to be turned in): 3.2 problems 1-34, 36, 37, 39, 41, 43, 45.
Homework 4 (due Tuesday, February 12). Solutions.
Thursday, February 7
3.3. Dimension. Rank-nullity theorem.
Tuesday, February 12
4.1. Linear spaces. 4.2 (up to Example 1). Linear transformations.
Homework 5 (due Tuesday, February 19). Solutions.
Thursday, February 14
4.2 (contd.) Isomorphisms. 4.3 (up to Example 3). The matrix of a linear transormation.
Tuesday, February 19
4.3 (contd.) Change of basis. 5.1 (up to Theorem 5.1.3). Orthogonality. Orthonormal bases.
Homework 6 (due Tuesday, February 26). Solutions.
Thursday, February 28
Review.
Homework 7 (due Tuesday, March 12). Solutions.
Tuesday, March 12: Midterm Exam. Solutions.
Covers Chapters 1 - 4. Books, notes, or calculators are not allowed. You can bring one letter-size handwritten sheet. Only write on one side of that sheet. No photocopies of anything are allowed.
Below are some helpful materials from previous years. When you use them, ignore the material that we have not covered.
Review Sheet 1 and Review Sheet 2.
Sample Midterm 1, Sample Midterm 2, Sample Midterm 3, Sample Midterm 4, Sample Midterm 5.
Most importantly, review the homework problems (and solutions posted above), and solve as many exercices from the textbook as possible.
Homework 8 (due Tuesday, March 19). Solutions.
Thursday, March 14
5.3. Orthogonal matrices. 5.4. Least squares.
Tuesday, March 19
6.1, 6.2 (begins). Determinants.
Homework 9 (due Tuesday, March 26). Solutions.
Thursday, March 21
6.2 (end), 6.3. Determinants and Gaussian elimination. Cramer's rule.
Tuesday, March 26
7.1, 7.2 (begins). Eigenvalues, eigenvectors.
Homework 10 (due Tuesday, April 2). Solutions.
Thursday, March 28
7.2, 7.3. Properties of eigenvalues. Eigenspaces. Multiplicities.
Tuesday, April 2
7.4. Diagonalization.
Homework 11 (due Tuesday, April 9). Solutions.
Thursday, April 4
7.5. Complex eigenvalues. Jordan canonical form (without proof). Complex vectors and matrices. Inner product. Lecture notes [some pages are by Dr. Wang].
Tuesday, April 9
8.1. Symmetric matrices. Spectral theorem. Spectral decomposition. Lecture notes.
Homework 12 (due Tuesday, April 16). Solutions.
Thursday, April 11
Tuesday, April 16
8.3. Singular values and singular vectors. Singular value decomposition.
Homework 13 (due Tuesday, April 23). Solutions.
Thursday, April 18
Finding rank, image, kernel, inverse from SVD. Pseudoinverse. Applications for solving linear systems and least-squares problems.
Tuesday, April 23
Review.
Wednesday, May 1: Final Exam. Section 1: 4:00 pm - 6:00 pm. Section 2: 1:30 - 3:30 pm. In the same room where the class regularly met.
Covers Chapters 5 - 8. Books, notes, or calculators are not allowed. You can bring one letter-size handwritten sheet. Only write on one side of that sheet. Photocopies are not allowed.
Below are some helpful materials from previous years. When you use them, ignore the material that we have not covered.
Review Sheet
Sample Final 1, Sample Final 2, Sample Final 3, Sample Final 4,
Most importantly, review the homework problems (and solutions posted above), and solve as many exercices from the textbook as possible.

Course webpage: http://www-personal.umich.edu/~romanv/teaching/2012-13/419/419.html