M531 Learning Objectives

- 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. - Develop the basic
**theory**of linear algebra, which is so important in systems theory and applied mathematics. - Provide an introduction into
**representations**and**approximations**within infinite-dimensional spaces. - Develop
**analytical**solutions of partial differential equations describing important physical processes, such as heat conduction and diffusion.

__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