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
https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf
Search by ISBN on Amazon: 9789814723770
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 L13; 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; 12.3.12 

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, LUfactorization 
Covers L912; 12.3.34 

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 
L1518; 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 
L1921 

22 
CauchySchwarz, triangle inequality, Pythagoras 
Chapter 9.3 
23 
Orthonormal bases, GramSchmidt procedure 
Chapter 9.4 – 9.5 
24 
Orthogonal projections, minimization problems 
Chapter 9.6 
Discussion: GramSchmidt procedure and orthogonal projections 
L2224 

25 
Change of bases 
Chapter 10 
26 
Selfadjoint and normal operators 
Chapter 11.1 – 11.2 
27 
Spectral theorem for normal maps (complex) 
Chapter 11.3 
Discussion: Change of basis, diagonalization 
L2527 

28 
Diagonalization 
Chapter 11.4 
29 
Positive operators, polar and singular value decompositions 
Chapter 11.6 – 11.7 
https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf