Geometric Algebra

Emil Artin

• Publisher: Dover Publications
• Year: 2016
• Print ISBN: 9780486801551
• eBook ISBN: 9780486809205
• Topic: Mathematics
• Subtopic: Linear Algebra
• Index: Mathematics

This concise classic presents advanced undergraduates and graduate students in mathematics with an overview of geometric algebra. The text originated with lecture notes from a New York University course taught by Emil Artin, one of the preeminent mathematicians of the twentieth century. The Bulletin of the American Mathematical Society praised Geometric Algebra upon its initial publication, noting that "mathematicians will find on many pages ample evidence of the author's ability to penetrate a subject and to present material in a particularly elegant manner."
Chapter 1 serves as reference, consisting of the proofs of certain isolated algebraic theorems. Subsequent chapters explore affine and projective geometry, symplectic and orthogonal geometry, the general linear group, and the structure of symplectic and orthogonal groups. The author offers suggestions for the use of this book, which concludes with a bibliography and index.

CHAPTER I

Preliminary Notions

1. Notions of set theory

We begin with a list of the customary symbols:

a ε S	means a is an element of the set S.
S ⊂ T	means S is a subset of T.
S ∩ T	means the intersection of the sets S and T; should it be empty we call the sets disjoint.
S ∪ T	stands for the union of S and T.

∩_i S_i and ∪_i S_i stand for intersection and union of a family of indexed sets. Should S_i and S_i be disjoint for i ≠ j we call ∪_i S_i a disjoint union of sets. Sets are sometimes defined by a symbol {···} where the elements are enumerated between the parenthesis or by a symbol {x|A} where A is a property required of x; this symbol is read: “the set of all x with the property A”. Thus, for example:

If f is a map of a non-empty set S into a set T, i.e., a function f(s) defined for all elements s ε S with values in T, then we write either

If

and

we also write

. If s ε S then we can form g(f(s)) ε U and thus obtain a map from S to U denoted by

. Notice that the associative law holds trivially for these “products” of maps. The order of the two factors gf comes from the notation f(s) for the image of the elements. Had we written (s)f instead of f(s), it would have been natural to write fg instead of gf. Although we will stick (with rare exceptions) to the notation f(s) the reader should be able to do everything in the reversed notation. Sometimes it is even convenient to write s^f instead of f(s) and we should notice that in this notation (s^f)^g = s^gf.

If

and S₀ ⊂ S then the set of all images of elements of S₀ is denoted by f(S₀); it is called the image of S₀. This can be done particularly for S itself. Then f(S) ⊂ T; should f(S) = T we call the map onto and say that f maps S onto T.

Let T₀ be a subset of T. The set of all s ε S for which f(s) ε T₀ is called the inverse image of T₀ and is denoted by f⁻¹(T₀). Notice that f⁻¹(T₀) may very well be empty, even if T₀ is not empty. Remember also that f⁻¹ is not a map. By f⁻¹(t) for a certain t ε T we mean the inverse image of the set {t} with the one element t. It may happen that f⁻¹(t) never contains more than one element. Then we say that f is a one-to-one into map. If f is onto and one-to-one into, then we say that f is one-to-one onto, or a “one-to-one correspondence.” In this case only can f⁻¹ be interpreted as a map

and is also one-to-one onto. Notice that f⁻¹f : S → S and ff⁻¹ : T → T and ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• Suggestions for The Use of this Book
• Chapter I: Preliminary Notions
• Chapter II: Affine and Projective Geometry
• Chapter III: Symplectic and Orthogonal Geometry
• Chapter IV: The General Linear Group
• Chapter V: The Structure of Symplectic and Orthogonal Groups
• Bibliography
• Index