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Applied Linear Algebra 1

This is the courseware for MATH 106: Applied Linear Algebra 1 at the University of Waterloo.

This course explores systems of linear equations, matrix algebra, determinants, and introduces vector spaces and their applications.

Euclidean Vector Spaces

In this module, we will introduce you to the concept of a vector and some of the fundamental ideas of linear algebra: vector addition, scalar multiplication, span, and linear independence. We will also explore the topics of dot products and cross products, which only apply to the Euclidean vectors we will study in this module.

Systems of Linear Equations

In this module, we will develop an algorithm to solve systems of linear equations so that we can answer the questions about span and linear independence posed previously.

Matrices, Linear Mappings, and Inverses

In this module, we begin a vector-like study of the matrices introduced previously. We will define matrix addition, scalar multiplication, span and linear independence. Unlike the Euclidean vectors, we will also define matrix multiplication and matrix inverses. We will also use matrices to explore linear mappings, another fundamental concept in linear algebra.


In this module, we continue our study of matrices by introducing a new idea: the determinant of a matrix. We will learn how to calculate the determinant and how to use it to answer standard linear algebra questions.

Eigenvectors and Diagonalization

In this module, we undertake a brief study of eigenvalues and eigenvectors. We learn how to find them and how to use them to diagonalize a matrix.