MATH 3273 Linear Algebra II
Students who take this course will require knowledge of the basic and some advanced concepts of Algebra. Thus, ABSTRACT ALGEBRA I & II and LINEAR ALGEBRA I are both listed as prerequisites.
The first part of the course continues the treatment of Vector Spaces and Linear Transformations started in LINEAR ALGEBRA I. The Rank-Nullity Theorem is stated and proved. Linear transformations are then viewed as elements of a larger algebraic structure, the algebra. In this formal context, the idea of polynomials of linear transformations is developed.
The theory of eigenvalues and eigenvectors is fundamental to Linear Algebra, and the course proceeds to study the same in detail. The connection between polynomials of matrices and their eigenvalues is explored and the celebrated CayleyHamilton Theorem is proved.
At this point, the students become aware that an algorithm for writing a matrix in a standard form, where the eigenvalues of the matrix may be easily obtained, is desirable. With this motivation, the existence and uniqueness of the Jordan Normal Form is proved. Techniques for computing the Jordan Normal Form are presented. The applications and limitations of the Jordan Normal Form are discussed.
The module is a natural generalisation of a vector space, and any student of advanced Linear Algebra should be familiar with the structure. The course therefore proceeds to define the module, giving motivating examples. The fundamental theorems are proved, drawing parallels with the algebraic structures which the students have already met. The existence and uniqueness of the Rational Canonical Form are stated here. Proofs may be sketched, but are not examinable.
The course then turns to vector spaces over the complex numbers, where the concept of an inner product is introduced. The properties of the inner product are discussed, and the fundamental definitions of unitary and Hermitian (in the context of linear transformations and matrices) are made. The base field is then further restricted to the reals, and the results developed are specialised to this case. An elegant proof of the Spectral Theorem for real symmetric matrices is given. The material developed here is applied to the study of quadratic forms.
The true power of Linear Algebra lies in its adaptability to computational tasks. As an illustration of this, the Singular Value Decomposition is introduced and its applications are discussed.
Traditionally, the tools of Linear Algebra have been heavily used in geometrical applications. As a demonstration of this, the material developed on quadratic forms is used to investigate the nature of quadric surfaces.
Since cogent communication of mathematical ideas is important in the presentation of proofs, the course will emphasize clear, concise exposition. This course will therefore be useful for all students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics in mathematics.