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Chapter1Vectors, the Dot Product, and Matrices

This first chapter introduces the basic objects of linear algebra. You will meet vectors, the dot product, matrices, and the computational system SageMath.

Vectors are a generalization of the concept of number. Where real numbers can help us model the geometry of points on a line, vectors will allow us to model the geometry of a plane, or (three-dimensional) space, or even "spaces" with higher dimensions. We begin by learning about the algebra of vectors, and making connections to the geometry.

A running theme for this course is the use of the SageMath mathematical software system. In order to get started, the second section of this course is dedicated to getting started with SageMath using the SageMathCloud (SMC). You will make an account and run through an introductory workshop with SMC. Also, throughout this workbook you will find little embedded pieces of SageMath code. These are implemented using the SageMath SingleCell server. Most of these SageMath cells are mutable. You can change the content in them and re-evaluate your new SageMath code. I encourage you to play with these---it is a good way to learn the basics of SageMath.

The dot product is a funny kind of multiplication. It plays an important role in mathematics because it captures all of the basics of measurement. We shall learn how to use the dot product to measure lengths and angles. By its definition, the dot product is connected with the concept of a linear equation, so it will make frequent appearances in our work.

Matrices are another way to generalize the concept of number. (In fact, they generalize the concept of vector.) We start here by learning about the algebra of matrices. The whole rest of this course will focus on matrices, their uses, and their properties.

The fifth and final section of the chapter is a short assignment designed to consolidate learning. You will get a chance to practice your skills and to think more deeply about the concepts you have learned.