Preface-6

Applications in the Book and on the Website

I hope this book will be useful to you long after the linear algebra course is complete. It is all the applications of linear algebra that make this possible. Matrices carry data, and other matrices operate on that data. The goal is to "see into a matrix" by understanding its eigenvalues and eigenvectors and singular values and singular vectors. And each application has special matrices—here are four examples:

Markov matrices M Each column is a set of probabilities adding to 1. Incidence matrices A Graphs and networks start with a set of nodes. The matrix A tells the connections (edges) between those nodes. Transform matrices F The Fourier matrix uncovers the frequencies in the data. Covariance matrices C The variance is key information about a random variable. The covariance explains dependence between variables.

We included those applications and more in this Sixth Edition. For the crucial computation of matrix weights in deep learning, Chapter 9 presents the ideas of optimization. This is where linear algebra meets calculus: derivative = zero becomes a matrix equation at the minimum point because $F(x)$ has many variables.

Several topics from the Fifth Edition gave up their places but not their importance. Those sections simply moved onto the Web. The website for this new Sixth Edition is math.mit.edu/linearalgebra

That website includes sample sections from this new edition and solutions to all Problem Sets. These sections (and more) are saved online from the Fifth Edition:

Fourier Series Norms and Condition Numbers Iterative Methods and Preconditioners Linear Algebra for Cryptography

Here is a small touch of linear algebra—three questions before this course gets serious:

1. Suppose you draw three straight line segments of lengths $r$ and $s$ and $t$ on this page. What are the conditions on those three lengths to allow you to make the segments into a triangle? In this question you can choose the directions of the three lines.

2. Now suppose the directions of three straight lines $u$, $v$, $w$ are fixed and different. But you could stretch those lines to $au$, $bv$, $cw$ with any numbers $a, b, c$. Can you always make a closed triangle out of the three vectors $au, bv, cw$?

3. Linear algebra doesn't stay in a plane! Suppose you have four lines $u, v, w, z$ in different directions in 3-dimensional space. Can you always choose the numbers $a, b, c, d$ (zeros not allowed) so that $au + bv + cw + dz = 0$?

For typesetting this book, maintaining its website, offering quality textbooks to Indian fans, I am grateful to Ashley C. Fernandes of Wellesley Publishers (www.wellesleypublishers.com)

gilstrang@gmail.com

Gilbert Strang