# Introduction to Linear Algebra, Fifth Edition

## Book Preface

I am happy for you to see this Fifth Edition of Introduction to Linear Algebra. This is the text for my video lectures on MIT’s OpenCourseWare (ocw.mit.edu and also YouTube). I hope those lectures will be useful to you (maybe even enjoyable!). Hundreds of collges and universities have chosen this textbook for their basic linear algebra course. A sabbatical gave me a chance to prepare two new chapters about probability and statistics and understanding data. Thousands of other improvements tooprobably only noticed by the author. . . Here is a new addition for students and all readers: Every section opens with a brief summary to explain its contents. When you read a new section, and when you revisit a section to review and organize it in your mind, those lines are a quick guide and an aid to memory.

Another big change comes on this book’s website math.mit.edu/linearalgebra. That site now contains solutions to the Problem Sets in the book. With unlimited space, this is much more flexible than printing short solutions. There are three key websites : ocw.mit.edu Messages come from thousands of students and faculty about linear algebra on this OpenCourseWare site. The 18.06 and 18.06 SC courses include video lectures of a complete semester of classes. Those lectures offer an independent review of the whole subject based on this textbook-the professor’s time stays free and the student’s time can be 2 a.m. (The reader doesn’t have to be in a class at all.) Six million viewers around the world have seen these videos (amazing). I hope you find them helpful. web.mit.edu/18.06 This site has homeworks and exams (with solutions) for the current course as it is taught, and as far back as 1996. There are also review questions, Java demos, Teaching Codes, and short essays (and the video lectures). My goal is to make this book as useful to you as possible, with all the course material we can provide. math.mit.edu/linearalgebra This has become an active website. It now has Solutions to Exercises-with space to explain ideas. There are also new exercises from many different sources-practice problems, development of textbook examples, codes in MATLAB and Julia and Python, plus whole collections of exams (18.06 and others) for review. Please visit this linear algebra site. Send suggestions to [email protected]

The cover shows the Four Fundamental Subspaces-the row space and nullspace are on the left side, the column space and the nulls pace of AT are on the right. It is not usual to put the central ideas of the subject on display like this! When you meet those four spaces in Chapter 3, you will understand why that picture is so central to linear algebra. Those were named the Four Fundamental Subspaces in my first book, and they start from a matrix A. Each row of A is a vector in n-dimensional space. When the matrix has m rows, each column is a vector in m-dimensional space. The crucial operation in linear algebra is to take linear combinations of column vectors. This is exactly the result of a matrix-vector multiplication. Ax is a combination of the columns of A.

When we take all combinations Ax of the column vectors, we get the column space. If this space includes the vector b, we can solve the equation Ax = b. May I call special attention to Section 1.3, where these ideas come early-with two specific examples. You are not expected to catch every detail of vector spaces in one day! But you will see the first matrices in the book, and a picture of their column spaces.

There is even an inverse matrix and its connection to calculus. You will be learning the language of linear algebra in the best and most efficient way: by using it. Every section of the basic course ends with a large collection of review problems. They ask you to use the ideas in that section–the dimension of the column space, a basis for that space, the rank and inverse and determinant and eigenvalues of A. Many problems look for computations by hand on a small matrix, and they have been highly praised. The Challenge Problems go a step further, and sometimes deeper. Let me give four examples:

Section 2.1: Which row exchanges of a Sudoku matrix produce another Sudoku matrix?

Section 2.7: If Pis a permutation matrix, why is some power pk equal to I?

Section 3.4: If Ax= band Cx = b have the same solutions for every b, does A equal C?

Section 4.1: What conditions on the four vectors r, n, c, £ allow them to be bases for

the row space, the nullspace, the column space, and the left nullspace of a 2 by 2 matrix?

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