MATH 341: Linear Algebra and Matrix Theory

General Information

Instructor: Oscar Gonzalez (og@math.utexas.edu) 471-1138

Office hrs: RLM 10.122, W 4-5pm, Th 8-9:30am

Text : Elementary Linear Algebra, by Andrilli and Hecker, 3rd Edition (2003, Elsevier)

Lecture : RLM 6.126, MWF 9-10am

Course webpage: http://www.ma.utexas.edu/~og/TEACHING/M341/m341.html

Description

This course covers a variety of topics in the theory of vectors and matrices and provides an introduction to proofs and abstract mathematics. The course is aimed at students in the mathematical sciences and its objective is to expose students to the basic theory of linear algbera and develop their proof-writing skills. The main topics to be covered as time permits are vectors and matrices, systems of linear equations, determinants and eigenvalues, vector spaces, linear transformations and orthogonality.

Prerequisites

M408D with a grade of C or better.

Policies

Homeworks: Homework sets will be assigned throughout the semester (about every week or so) and will contain a mix of computational and analytical problems. It is acceptable for students to discuss the homework sets with each other; however, each student must write up and submit their own work. Only a selected portion of the assigned problems may be graded on any given homework set.


Technology: Various homework problems in the course involve tedious computations. To reduce the amount of such computations, students will be allowed to use a calculator or a mathematical software package to complete certain homework problems. Computers can be found in the Undergraduate Computer Lab, RLM 7.122.


Exams: There will be 3 exams (the third exam to be held during the time scheduled by the University for the Final Examination). The tentative dates are

Exam #1 (Oct 7), Exam #2 (Nov 9), Exam #3 (Dec 14, 7pm).


Course grade: Your overall course grade will be determined by your homework and exam grades in the following proportions

Homeworks (20%), Exam #1 (25%), Exam #2 (25%), Exam #3 (30%).


Grade scale: If necessary, exam grades will be curved to achieve a class average equivalent to 75/100 points or thereabouts. Grades will never be curved downwards. The overall course grade will be based on a standard grading scale (A=100-90%, B=89-80%, C=79-70%, D=69-60%, F=59-0%).


Students with disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.

Tentative schedule

The following table is a tentative schedule for the course. Please be aware that material may be reordered, added or deleted.
1. Vectors and matrices (2 weeks)

• Vector addition, scaling, dot product
  
• Cauchy-Schwarz and Triangle inequalities
  
• Matrix addition, scaling, multiplication
  
• Proof techniques
  

2. Systems of linear equations (2 weeks)

• Solution sets, Gauss elimination
  
• Consistent and inconsistent systems
  
• Row echelon forms, Gauss-Jordan reduction
  
• Rank, row space and inverse of a matrix
  

3. Determinants and eigenvalues (2 weeks)

• Determinant of a matrix, properties
  
• Cofactors and adjoint of a matrix
  
• Matrix inverse formula, Cramer's rule
  
• Eigenvalues and eigenvectors
  
• Diagonal factorization of a matrix
  

4. Vector spaces (3 weeks)
• Idea of vector spaces, subspaces
  
• Span and linear independence of vectors
  
• Bases and dimension of vector spaces
  
• Coordinates with respect to a basis
  
• Change of basis formula for vectors
  

5. Linear transformations (2 weeks)
• Idea of a linear transformation
  
• Matrix representation
  
• Kernel and range, Dimension Theorem
  
• Eigenvalues, eigenvectors and diagonalization
  

6. Orthogonality (2 weeks)
• Orthonormal bases, Gram-Schmidt process
  
• Orthogonal complements, Projection Theorem
  
• Orthogonal projections of vectors
  
• Diagonalization of symmetric operators