# 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
https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf
Search by ISBN on Amazon: 9789814723770
Prerequisites:
MAT 021C C- or better or MAT 021CH C- or better.
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