Course Credit: 3
Linear algebra II is the study of vector spaces and linear mappings between them. In this course, we will begin by reviewing topics you learned in Linear Algebra I, starting with vectors, matrices and linear mappings. The review will refresh the student's knowledge of the fundamentals of vectors and of matrix theory, and how to perform operations on matrices. After the review, we can extend this idea to Similar Matrices. Next, we will focus on Linear Functional and dual Space. We will then introduce a new structure on vector spaces: an inner product. Inner products allow us to introduce geometric aspects, such as length of a vector, and to define the notion of orthogonality between vectors. In this context, we will study the applications in Linear Models and Fourier Approximation, and more. We will end this chapter with the spectral theorem, which provides a decomposition of the vector space on which operators act, and singular-value decomposition, which is a generalization of the spectral theorem to arbitrary matrices. Then, we will study Bilinear, quadratic & hermitian forms. Symmetric Matrices and Quadratic Forms, Positive Definite Matrices will be studied at the end of this course with their applications in diverse fields. The subject material is of vital importance in all fields of mathematics and in science in general.