About MAS 3105

MAS 3105 Matrix and Linear Algebra is what is commonly called a first course in linear algebra for undergraduate students. The intended audience is primarily for mathematics, statistics, science, engineering, and education majors.

The course is primarily but not exclusively a computational course, in the sense that many of our explanations and examples will be computational, and students will be expected to show mastery by solving problems. Nonetheless, there is a certain amount of mathematical proof of relatively abstract theorems; most of this will come in the audio/video lectures, and students may be required to write simple proofs or justifications of some statements.

Generally speaking, linear algebra is the study of solutions of systems of linear equations. Of course, it goes much further than that, but the most fundamental question is: how do we find (or express or predict or measure or ...) the solutions to a system of linear equations? As you'll see, 95% or more of what we do in this class will come back to solving a related system of linear equations.

Along the way, we develop the basic language and arithmetic of matrices (and associated linear transformations), and late in the course will talk about the important concepts of eigenvalues and eigenvectors of matrices and what they tell us about the matrix (and by extension, about the particular context in statistics, physics, engineering, etc.)

It is not an overstatement to say that along with Calculus and Differential Equations, Linear Algebra is used in nearly every scientific field.

Course Objectives

To learn the basic mathematics of linear algebra, including:
- how to effectively solve systems of linear equations
- how to represent the solution set of a system of linear equation, and how to use the representation in a wide variety of more advanced settings
- the arithmetic of matrices, including matrix inverses and determinants
- the eigenvalues and eigenvectors of a matrix, as one of the most important invariants associated to a matrix
To learn the essentials of vector spaces over the real numbers
- the primary examples of spaces of column vectors, of polynomials, and of matrices, among others
- the important ideas of linear combinations, spanning, and independence, and the implications they have for bases of vector spaces and hence for the dimension of a space
- the fundamental subspaces associated to a matrix: the null space, the column space, and the row space
- inner product spaces and the important notion of orthogonality
To combine and synthesize how systems of linear equations and vector spaces interact
- via the fundamental subspaces associated to a matrix
- via the eigenspaces associated to a matrix, and hence questions about diagonalizability
- via the use of orthogonality leading to the very important diagonalization of symmetric matrices and then to the singular value decomposition
Along the way, to learn some basic applications.