# Matrix Algebra

Massive Open Online Course
• Overview
• Course Content
• Requirements & Materials
Overview

## Matrix Algebra

Course Description

Your ability to apply the concepts that we introduced in our previous course is enhanced when you can perform algebraic operations with matrices. At the start of this class, you will see how we can apply the Invertible Matrix Theorem to describe how a square matrix might be used to solve linear equations. This theorem is a fundamental role in linear algebra, as it synthesizes many of the concepts introduced in the first course into one succinct concept.

You will then explore theorems and algorithms that will allow you to apply linear algebra in ways that involve two or more matrices. You will examine partitioned matrices and matrix factorizations, which appear in most modern uses of linear algebra. You will also explore two applications of matrix algebra, to economics and to computer graphics.

Learners taking this class are encouraged to first complete the first course in this series, Linear Equations.

Course Content

MATRIX OPERATIONS

INVERSE OF A MATRIX

INVERTIBLE MATRICES

APPLICATION: PARTITIONED MATRICES

APPLICATION: THE LU FACTORIZATION

APPLICATION: THE LEONTIF INPUT-OUTPUT MODEL

COMPUTER GRAPHICS

SUBSPACES OF RN

DIMENSION AND RANK

Requirements & Materials
Prerequisites

Recommended

• High school algebra, geometry, and pre-calculus

Required

• Linear Equations (DL 0050M)

Materials

Required

• Internet connection (DSL, LAN, or cable connection desirable)

### Who Should Attend

This course is designed for undergraduate students, advanced high school students, who are interested in pursuing any career path or degree program that involves linear algebra, or industry employees who are seeking a better understanding of linear algebra for their career development. ### What You Will Learn

• The invertibility of a matrix using the Invertible Matrix Theorem
• An LU factorization of a matrix and apply the LU factorization to solve systems of equations
• Transformation matrices construction to represent composite transforms with homogeneous coordinates, which are needed for computer graphics applications
• How to construct a basis for a subspace
• How to calculate the coordinates of a vector in a given basis
• How to characterize a matrix using the concepts of rank, column space, and null space ### How You Will Benefit

• Apply matrix algebra, the matrix transpose, and the zero and identity matrices, to solve and analyze matrix equations.
• Apply the formal definition of an inverse, and its algebraic properties, to solve and analyze linear systems.
• Apply partitioned matrices to solve problems regarding matrix invertibility and matrix multiplication.
• Apply the Rank, Basis, and Matrix Invertibility theorems to describe matrices, subspaces, and systems.
• Apply matrix algebra concepts to model, solve, and analyze real-world situations, including the Leontif Input-Output Model and computer graphics.
• Construct, or give examples of, mathematical expressions that involve vectors, matrices, and linear systems of linear equations.
• Analyze mathematical statements and expressions involving linear systems and matrices. For example, to describe solutions of systems in terms of invertibility and subspaces.
• ##### Taught by Experts in the Field
•  - Abe Kani
President