# Math 413 Advanced Linear Algebra

(3:3:0)

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

1. Linear equations (row operations, matrix multiplication, and invertibility)
2. Vector spaces
3. Linear transformations (algebra of linear transformations, isomorphisms, linear functionals, duality)
4. Polynomials and determinants (algebra of polynomials, polynomial ideals, determinant functions, permutations and uniqueness of determinants)
5. Jordan canonical form and elementary canonical forms (invariant subspaces, simultaneous diagonalization and triangulation, direct-sum decompositions, rational forms, Jordan form)
6. 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):