Preface
I will be happy with this preface
if
three important points come through clearly:
1. The beauty and variety
of
linear algebra, and its extreme usefulness
2. The goals
of
this book, and the new features in this Fourth Edition
3. The steady support from our linear algebra websites and the video lectures
May I begin with notes about two websites that are constantly used, and the new one.
ocw.mit.edu Messages come from thousands
of
students and faculty about linear algebra
on this OpenCourseWare site. The
18.06 course includes 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 3 a.m.
(The reader doesn't have to be in a class at all.) A 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 as possible, with all the course material we can provide.
math.mit.edu/linearalgebra The newest website is devoted specifically
to
this Fourth Edi-
tion. It will be a permanent record
of
ideas and codes and good problems and solutions.
Sevetal sections
of
the book are directly available online, plus notes on teaching linear
algebra. The content is growing quickly and contributions are welcome from everyone.
The Fourth Edition
Thousands
of
readers know earlier editions
of
Introduction
to
Linear Algebra. The new
cover shows the
Four
Fundamental
SUbspaces-the
row space and nullspace are on
the left side, the column space and the nullspace
of
AT
are on the right. It is not usual
to put the central ideas
of
the subject on display like this!
You
will meet those four spaces
in Chapter 3, and 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
v
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