Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 67: Modern Linear Algebra
Approved: 2015-05-01 (revised 2011-07-01, )

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
“Linear Algebra as an Introduction to Abstract Mathematics” by Isaiah Lankham, Bruno Nachtergaele and Anne Schilling
http://www.math.ucdavis.edu/%7Eanne/mat67_course_notes.pdf

Suggested Schedule:

Lec #

Topics

Comments/Topics

1

What is linear algebra?

Chapter 1

2, 3

Complex numbers

Chapter 2


Discussion: Calculations with complex numbers; encoding linear systems

Covers L1-3; 12.1

4

Fundamental theorem of algebra (proof optional)

Chapter 3

5

Vector spaces and subspaces

Chapter 4.1 - 4.3

6

Direct sum, linear span

Chapter 4.4 – 5.1


Discussion: Vector space of matrices and operations on matrices

Covers L5, 6; 12.2

7

Linear independence of vectors

Chapter 5.2

8

Bases and dimensions of vector spaces

Chapter 5.3 – 5.4

9

Linear maps

Chapter 6.1


Discussion: Linear independence, homogenous linear systems, Gaussian elimination

Covers L7, 8; 12.3.1-2

10

Null space and range of linear maps

Chapter 6.2 – 6.4

11

Dimension formula for a linear map

Chapter 6.5

12

Matrix of a linear map

Chapter 6.6


Discussion: Linear maps, inhomogeneous systems, LU-factorization

Covers L9-12; 12.3.3-4

13

Invertibility

Chapter 6.7

14

Midterm


15

Eigenvalues and eigenvectors

Chapter 7.1 – 7.3


Discussion: Linear maps

L12, 13; 12.4

16

Existence of eigenvalues

Chapter 7.4

17

Upper triangular matrix representation

Chapter 7.5

18

Diagonalization (2x2) and applications

Chapter 7.6


Discussion: Eigenvalues and eigenvectors, special operations on matrices

L15-18; 12.5

19

Permutations and the determinant

Chapter 8.1 – 8.5

20

Properties of the determinant

Chapter 8.6 – 8.7

21

Inner product spaces

Chapter 9.1 – 9.2


Discussion: Calculation of the determinant, inner product spaces

L19-21

22

Cauchy-Schwarz, triangle inequality, Pythagoras

Chapter 9.3

23

Orthonormal bases, Gram-Schmidt procedure

Chapter 9.4 – 9.5

24

Orthogonal projections, minimization problems

Chapter 9.6


Discussion: Gram-Schmidt procedure and orthogonal projections

L22-24

25

Change of bases

Chapter 10

26

Self-adjoint and normal operators

Chapter 11.1 – 11.2

27

Spectral theorem for normal maps (complex)

Chapter 11.3


Discussion: Change of basis, diagonalization

L25-27

28

Diagonalization

Chapter 11.4

29

Positive operators, polar and singular value decompositions

Chapter 11.6 – 11.7

Additional Notes:

Lecture notes “Linear Algebra as an Introduction to Abstract Mathematics by Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling are available on the class website at:

http://www.math.ucdavis.edu/~anne/mat67_course_notes.pdf

Learning Goals:

A goal of this course is to ensure students learn to write rigorous proofs and how to communicate mathematical concepts using language. Have students regularly practice writing formal proofs that emphasize course content and mathematical thinking using language.