M531 Learning Objectives

Overall Course Objectives

1. Develop a strong foundation in linear systems and maps, and analytical mathematics that will provide a basis for advanced studies in engineering, physics, and mathematics.
2. Develop the basic theory of linear algebra, which is so important in systems theory and applied mathematics.
3. Provide an introduction into representations and approximations within infinite-dimensional spaces.
4. Develop analytical solutions of partial differential equations describing important physical processes, such as heat conduction and diffusion.

Part I. Linear Algebra Learning Objectives

Overview: This part comprises approx. 40% of the course. The section is preceded with some background on mathematical modeling and some applications. The text (Axler, Linear Algebra Done Right, Springer, 1997) has a desirable order of presentation, although it seems a little poor in elementary examples, in particular on matrices as linear maps. It is dealing with the abstract part up front, and dealing with linear transformation and (complex) eigenvalues/eigenvectors in a rigorous way.

Getting the "more abstract point of view" is one of the strongest objectives of this part; this is not an undergraduate linear algebra course and it is very important for the students they understand how the properties (or theorems) are in fact established, and to see that it is not above their capabilities.

The text is generally followed, with the following considerations: complex spaces are used; difficult technical proofs are omitted, only some proofs that are useful in applications or that can offer more insight on the concepts are done in class. The following chapters are omitted: Chapter 4, Chapters 6-9.

Section One: Vector Spaces:

a) Determine whether sets form vector spaces

b) Determine whether sets form subspaces

c) Determine whether vectors are independent

d) Determine basis vectors and identify dimension of vector spaces

e) Determine whether vector spaces are isomorphic

f) Determine rank of matrices

g) Learn the basic definitions arising in the description of vector spaces

h) Learn the basic approaches used to prove assertions

Section Two: Linear Maps

a) Determine whether a given function is a linear map

b) Determine the null space of a linear map

c) Determine the range of a linear map

d) Determine whether a linear map is injective and surjective

e) Determine whether a linear map is invertible

f) Determine the matrix representing a linear map

Section Three: Determinants

a) Compute determinants of matrices;

b) Expand using cofactors;

Section Four: Eigenvalues and Eigenvectors

a) Perform complex arithmetics

b) Understand the generalization of previous knowledge to systems with complex eigenvalues

c) Compute eigenvalues and eigenvectors of operators

d) Know when matrices can be diagonalized