Geometric linear algebra

881 Pages·2005·3.67 MB·English

by  I-Hsiung Lin

G
EOME TRIC
LINEAR ALGEBRA
Volume1
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GEOMETRIC
LINEAR ALGEBRA
Volume 1
I-Hsiung Lin
National Taiwan Normal University, China
WeWorld Scientific
NEW JERSEY · LONDON · SINGAPORE · BEIJING · SHANGHAI · HONG KONG · TAIPEI · CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Lin, Yixiong.
Geometric linear algebra / I-hsiung Lin.
p. cm.
Includes bibliographical references and indexes.
ISBN 981-256-087-4 (v. 1) -- ISBN 981-256-132-3 (v. 1 : pbk.)
1. Algebras, Linear--Textbooks. 2. Geometry, Algebraic--Textbooks. I. Title.
QA184.2.L49 2005
512'.5--dc22
2005041722
British Library Cataloguing-in-Publication Data
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Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.
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July13,2005 17:31 WSPC/SPI-B244:GeometricLinearAlgebra—I fm
To the memory of my
grandparents, parents
and to my family
v
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July13,2005 17:31 WSPC/SPI-B244:GeometricLinearAlgebra—I fm
PREFACE
What is linear algebra about?
Many objects such as buildings, furniture, etc. in our physical world are
delicately constituted by counterparts of almost straight and flat shapes
which, in geometrical terminology, are portions of straight lines or planes.
A butcher’s customers ordered the meat in various quantities, some by
mass and some by price, so that he had to find answers to such questions
as: What is the cost of 2/3kg of the meat? What mass a customer should
haveifsheonlywantstospend10dollars?Thiscanbesolvedbythelinear
relationy =axory =ax+bwithbasabonus.Thesameis,whentraveling
abroad, to know the value of a foreign currency in term of one’s own. How
many faces does a polyhedron with 30 vertices and 50 edges have? What is
the Fahrenheit equivalent of 25◦C? One experiences numerous phenomena
in daily life, which can be put in the realms of straight lines or planes or
linear relations of several unknowns.
To start with, the most fundamental and essential ideas needed in
geometry are
1. (directed) line segment (including the extended line),
2. parallelogram (including the extended plane)
andtheassociatedquantitiessuchaslengthorsignedlengthoflinesegment
and angle between segments.
The algebraic equivalence, in global sense, is linear equations such as
a x +a x =b
11 1 21 2 1
or
a x +a x +a x =b
11 1 21 2 31 3 1
and simultaneous equations composed of them. The core is how to deter-
mine whether such linear equations have a solution or solutions, and if so,
how to find them in an effective way.
vii
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viii Preface
Algebrahasoperationalpriorityovergeometry,whilethelatterprovides
intuitively geometric motivation or interpretations to results of the former.
Both play a role of head and tail of a coin in many situations.
Linearalgebraisgoingtotransformtheafore-mentionedgeometricideas
intotwoalgebraicoperationssothatsolvinglinearequationscanbehandled
linearly and systematically. Its implication is far-reaching and its applica-
tion is widely-open and touches almost every field in modern science. More
precisely,
−(cid:1)
a directed line segment AB → a vector (cid:1)x;
ratio of signed lengths of (directed) line segments along the same line
−(cid:1)
PQ
−(cid:1) =α→(cid:1)y =α(cid:1)x,scalar multiplication of (cid:1)x by α.
AB
See Fig. P.1.
P
Q
B y
A
x
Fig.P.1
Hence, the whole line can be described algebraically as α(cid:1)x while α runs
through the real numbers. While, the parallelogram in Fig. P.2 indicates
−(cid:1) −(cid:1) −(cid:1)
that directed segments OA and BC represent the same vector (cid:1)x, OB and
−(cid:1)
AC represent the same vector (cid:1)y so that
−(cid:1)
the diagonal OC →(cid:1)x +(cid:1)y, the addition of vectors (cid:1)x and (cid:1)y.
C
B
y
y x+y
A
O x
Fig.P.2
As a consequence, the whole plane can be described algebraically as the
linear combinations α(cid:1)x +β(cid:1)y where α and β are taken from all the real
numbers. In fact, parallelograms provide implicitly as an inductive process
toconstructandvisualizehigherdimensionalspaces.Onemayimaginethe
line OA acting as an (n−1)-dimensional space, so that (cid:1)x is of the form
α1(cid:1)x1+···+αn−1(cid:1)xn−1. In case the point C is outside the space, (cid:1)y cannot
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Preface ix
be expressed as such linear combinations. Then the addition α(cid:1)y +(cid:1)x will
raise the space to the higher n-dimensional one.
As a whole, relying only on
α(cid:1)x, α∈R (real field) and
(cid:1)x +(cid:1)y
with appropriate operational properties and using the techniques:
linear combination,
linear dependence, and
linear independence
of vectors, plus deductive and inductive methods, one can develop and
establish the whole theory of Linear Algebra, even formally and in a very
abstract manner.
The main theme of the theory is about linear transformation which
can be characterized as the mapping that preserves the ratio of the signed
lengths of directed line segments along the same or parallel lines. Linear
transformations between finite-dimensional vector spaces can be expressed
asmatrixequations(cid:1)xA=(cid:1)y,afterchoosingsuitablecoordinatesystemsas
bases.
The matrix equation (cid:1)xA=(cid:1)y has two main features. The static struc-
ture of it, when consider (cid:1)y as a constant vector (cid:1)b, results from solving
algebraically the system (cid:1)xA = (cid:1)b of linear equations by the powerful and
usefulGaussianeliminationmethod.Rankofamatrixanditsfactorization
as a product of simpler ones are the most important results among all.
Rank provides insights into the geometric character of subspaces based on
the concepts of linear combination, dependence and independence. While
factorization makes the introduction of determinant easier and provides
preparatory tools to understand another feature of matrices. The dynamic
structure, when consider (cid:1)y as a varying vector, results from treating A as
a linear transformation defined by (cid:1)x →(cid:1)xA=(cid:1)y. The kernel (for homoge-
neous linear equations) and range (for non-homogeneous linear equations)
of a linear transformation, dimension theorem, invariant subspaces, diago-
nalizability,variousdecompositionsofspacesorlineartransformationsand
their canonical forms are the main topics among others.
When Euclidean concepts such as lengths and angles come into play, it
is the inner product that combines both and the Pythagorean Theorem or
orthogonality dominates everywhere. Therefore, linear operators (cid:1)y = (cid:1)xA