Linear Algebra

An Inquiry-Based Approach

Jeff Suzuki

• Publisher: CRC Press
• Year: 2021
• ISBN:
  9781000377514
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Linear Algebra: An Inquiry-based Approach is written to give instructors a tool to teach students to develop a mathematical concept from first principles. The Inquiry-based Approach is central to this development. The text is organized around and offers the standard topics expected in a first undergraduate course in linear algebra.

In our approach, students begin with a problem and develop the mathematics necessary to describe, solve, and generalize it. Thus students learn a vital skill for the 21st century: the ability to create a solution to a problem.

This text is offered to foster an environment that supports the creative process. The twin goals of this textbook are:

•Providing opportunities to be creative,
•Teaching "ways of thinking" that will make it easier for to be creative.

To motivate the development of the concepts and techniques of linear algebra, we include more than two hundred activities on a wide range of problems, from purely mathematical questions, through applications in biology, computer science, cryptography, and more.

Table of Contents

Introduction and Features
For the Student... and Teacher
Prerequisites
Suggested Sequences
1 Tuples and Vectors
2 Systems of Linear Equations
3 Transformations
4 Matrix Algebra
5 Vector Spaces
6 Determinants
7 Eigenvalues and Eigenvectors
8 Decomposition
9 Extras
Bibliography
Index

Bibliography

Jeff Suzuki is Associate Professor of Mathematics at Brooklyn College and holds a Ph.D. from Boston University. His research interests include mathematics education, history of mathematics, and the application of mathematics to society and technology. He is a two-time winner of the prestigious Carl B. Allendoerfer Award for expository writing. His publications have appeared in The College Mathematics Journals; Mathematics Magazine; Mathematics Teacher; and the American Mathematical Society's blog on teaching and learning mathematics. His YouTube channel (http://youtube.com/jeffsuzuki1) includes videos on mathematical subjects ranging
from elementary arithmetic to linear algebra, cryptography, and differential equations.

1

Tuples and Vectors

1.1 Tuples

We’ll begin our journey into linear algebra by considering a ubiquitous feature of life in the 21st century: personal information forms. For example, a medical record might look something like this:

Name: Jeff Suzuki	Gender: M
Height: 5’6”	Weight: 135
BP: 130 over 70	Age: 37

This form appears to contain information. But does it? In the 1940s, information theory pioneer Claude Shannon (1916–2001) made an important realization: If you already know the answer, the answer isn’t information. Thus the answer to “What is the person’s name?” is information: If we picked out a random form, we wouldn’t know the name of the person the form belongs to. But the answer to “What goes in the first box?” is not information: Regardless of which form we picked out, the same thing goes in the first box, so in some sense the label “Name” in the first box, as well as the other labels, are unnecessary.

To be sure, we couldn’t just use a blank form; we have to agree in advance that the name will go in the first box, the gender in the second, and so on. But once we make that agreement, it’s not necessary to keep the labels; we could simply record the answers to the questions:

$$\begin{array}{l}\text{Jeff}\text{Suzuki},\text{M},36,5^{\prime }{6}^{″},135,130\text{over}70\hfill \end{array}$$

This allows us to represent information spread over a sheet of paper into a compact list. In fa...