Linear Algebra is one of the central topics in mathematics today. It finds applications in probability, statistics, analysis, differential equations and a host of other areas, as well as being a topic of study in its own right.

In this course we will suppose that you have some fundamental computational skills with matrices and vectors, including matrix algebra, computation determinants, eigenvectors and eigenvalues, and manipulation of dot products. See this list of prerequisite topics for details. This material can be found in a host of places, including chapters 2-4 of Elementary Differential Equations with Linear Algebra by Albert Rabenstein, a standard textbook for Math 234 here at UWM.

Our goal is to prove the theorems hinted at in your earlier work and to expand your view of linear algebra as a tool in the rest of mathematics. Therefore, this course is not a course about computation, but rather about attacking the theory behind the calculations you have already mastered. By way of a parallel, this course stands in relation to your previous exposure to linear algebra topics as advanced calculus stands to your first courses in calculus.

Some of the things we will investigate are

• The solutions of simultaneous linear equations over arbitrary fields.
• The relation between linear transformations and matrices.
• A complete theory of determinants.
• How to determine if a matrix can be diagonalized.
• The generalization of dot products and their relation to symmetric matrices.
• What is the proper generalization of dot product.
• What is the proper generalization of symmetric matrices.

We will be reading Linear Algebra, Second Edition (ISBN 13-536797-2) by Kenneth Hoffman and Ray Kunze. This book is available from the publisher Prentice Hall , at the Union Bookstore, and at amazon.com. I will put a copy on reserve at the library if someone would like me to do so. It is a classic book on linear algebra, and well worth the price. I still refer to my copy that I bought back in 1974.

You will be reading the following sections of the text, but not in this order:

1. Linear Equations
   1. Fields
   2. Systems of Linear Equations
   3. Matrices and Elementary Row Operations
   4. Row-Reduced Echelon Matrices
   5. Matrix Multiplication
   6. Invertible Matrices
2. Vector Spaces
   1. Vector Spaces
   2. Subspaces
   3. Bases and Dimension
   4. Coordinates
   5. Summary of Row Equivalence
   6. Computations Concerning Subspaces
3. Linear Transformations
   1. Linear Transformations
   2. The Algebra of Linear Transformations
   3. Isomorphism
   4. Representation of Transformations by Matrices
   5. Linear Functionals
   6. Cross Products
   7. The Double Dual
   8. The Transpose of a Linear Transformation
4. Polynomials
   1. Algebras
   2. The Algebra of Polynomials
   3. Lagrange Interpolation
   4. Polynomial Ideals
   5. The Prime Factorization of a Polynomial
5. Determinants
   1. Commutative Rings
   2. Determinant Functions
   3. Permutations and Uniqueness of Determinants
   4. Additional Properties of Determinants
6. Elementary Canonical Forms I
   1. Introduction
   2. Inner Products
   3. Inner Product Spaces
   4. Linear Functionals and Adjoints
   5. Unitary Operators
   6. Normal Operators
7. Elementary Canonical Forms II
   1. Simultaneous Triangulation and Diagonalization
   2. Direct Sum Decomposition
   3. Invariant Direct Sums
   4. The Primary Decomposition Theorem

There will be weekly homework assignments (about 40% of your grade), a midterm exam (about 30% of your grade), and a final exam (about 30% of your grade). You will be required to re-submit homework problems until you have solved them correctly if you are to get credit for them.

If you have any doubts about your preparation, or any other questions, contact me .