Linear Algebra

What you Need to Know

Hugo J. Woerdeman

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000345896
• Edition: 1
• Topic: Matematica
• Subtopic: Algebra

There is good reason to be excited about Linear Algebra. With the world becoming increasingly digital, Linear Algebra is gaining more and more importance. When we send texts, share video, do internet searches, there are Linear Algebra algorithms in the background that make it work.

This concise introduction to Linear Algebra is authored by a leading researcher presents a book that covers all the requisite material for a first course on the topic in a more practical way.

The book focuses on the development of the mathematical theory and presents many applications to assist instructors and students to master the material and apply it to their areas of interest, whether it be to further their studies in mathematics, science, engineering, statistics, economics, or other disciplines.

Linear Algebra has very appealing features:
•It is a solid axiomatic based mathematical theory that is accessible to a large variety of students.
•It has a multitude of applications from many different fields, ranging from traditional science and engineering applications to more 'daily life' applications.
•It easily allows for numerical experimentation through the use of a variety of readily available software (both commercial and open source).

Several suggestions of different software are made. While MATLAB is certainly still a favorite choice, open-source programs such as Sage (especially among algebraists) and the Python libraries are increasingly popular. This text guides the student to try out different programs by providing specific commands.

1

Matrices and Vectors

CONTENTS

• 1.1 Matrices and Linear Systems
• 1.2 Row Reduction: Three Elementary Row Operations
• 1.3 Vectors in ℝⁿ, linear combinations and span
• 1.4 Matrix Vector Product and the Equation Ax = b
• 1.5 How to Check Your Work
• 1.6 Exercises

In this chapter we will introduce matrices and vectors in the context of systems of linear equations. Let us start with a system of two linear equations with two unknowns, such as

2x+3y=4,4x−y=−6.

To solve this system, one would take a combination of the two equations and eliminate one of the variables. We can take the second equation and subtract two times the first, giving us

(4x−y)−2(2x+3y)=−6−2·4whichsimplifiesto−​7y=−14,

yielding the equivalent system

2x+3y=4,−7y=−14.

Now we can solve for y (giv...