Topics in Linear Algebra

Department: 
MATH
Course Number: 
4305
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every Fall and Spring; Some Summers

Linear algebra in R^n, standard Euclidean inner product in R^n, general linear spaces, general inner product spaces, least squares, determinants, eigenvalues and eigenvectors, symmetric matrices.

Prerequisites: 

(MATH 1552 or MATH 1X52) and (MATH 1553 or MATH 1554 or MATH 1564 or MATH 1X53 or MATH 1X54)

Course Text: 

At the level of Linear Algebra with Applications, Bretscher

Topic Outline: 
  • Linear systems. Gauss-Jordan elimination (row reduction)
  • Linear transformations in R^n and their matrices
  • Composed transformations and matrix products. The inverse
  • Subspaces, bases, dimension, coordinates with respect to bases
  • Image and kernel. Rank and nullity
  • General linear spaces and subspaces
  • Linear transformations in general linear spaces. Matrices with respect to bases
  • Orthogonal projections
  • Orthonormal bases, Gram-Schmidt process, and QR factorization
  • Least squares
  • General Inner product spaces
  • Determinants. Geometric properties
  • Eigenvalues and eigenvectors
  • Diagonalization. Matrix iterations
  • Orthogonal diagonalization of symmetric matrices and quadratic forms
  • Singular value decomposition