# Math 413 Advanced Linear Algebra

## Contents

## Catalog Information

### Title

Advanced Linear Algebra

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

F and possibly W

### Prerequisite

Math 371 recommended, but not required.

### Description

Theory and advanced topics of linear algebra.

## Desired Learning Outcomes

**Prove:** Students will be able to prove central linear-algebraic results, as well as other results with similar derivations.

**Distinguish:** Students will be able to distinguish between true and plausibly-sounding false propositions in the language of linear algebra.

**Construct:** Students will be able to construct examples and counterexamples illustrating relations between different linear-algebraic concepts.

**Categorize:** Students will be able to categorize linear-algebraic structures according to their properties.

**Calculate:** Students will be able to calculate precisely and efficiently, choosing appropriate methods.

### Prerequisites

Math 371 recommended, but not required.

### Minimal learning outcomes

- Linear equations (row operations, matrix multiplication, and invertibility)
- Vector spaces
- Linear transformations (algebra of linear transformations, isomorphisms, linear functionals, duality)
- Polynomials and determinants (algebra of polynomials, polynomial ideals, determinant functions, permutations and uniqueness of determinants)
- Jordan canonical form and elementary canonical forms (invariant subspaces, simultaneous diagonalization and triangulation, direct-sum decompositions, rational forms, Jordan form)
- Inner product spaces (inner product spaces, linear functionals and adjoints, unitary operators, normal operators)

### Textbooks

Possible textbooks for this course include (but are not limited to):