Page 165 - Matrix Analysis & Applied Linear Algebra

160 Chapter 4 Vector Spaces

Vector Space Definition
The set V is called a vector space over F when the vector addition
and scalar multiplication operations satisfy the following properties.
(A1) x+y ∈V for all x, y ∈V. This is called the closure property
for vector addition.

(A2) (x + y)+ z = x +(y + z) for every x, y, z ∈V.
(A3) x + y = y + x for every x, y ∈V.
(A4) There is an element 0 ∈V such that x + 0 = x for every
x ∈V.
(A5) For each x ∈V, there is an element (−x) ∈V such that
x +(−x)= 0.
(M1) αx ∈V for all α ∈F and x ∈V. This is the closure
property for scalar multiplication.
(M2) (αβ)x = α(βx) for all α, β ∈F and every x ∈V.
(M3) α(x + y)= αx + αy for every α ∈F and all x, y ∈V.
(M4) (α + β)x = αx + βx for all α, β ∈F and every x ∈V.
(M5) 1x = x for every x ∈V.

A theoretical algebraic treatment of the subject would concentrate on the
logical consequences of these defining properties, but the objectives in this text
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are different, so we will not dwell on the axiomatic development. Neverthe-
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The idea of defining a vector space by using a set of abstract axioms was contained in a general
theory published in 1844 by Hermann Grassmann (1808–1887), a theologian and philosopher
from Stettin, Poland, who was a self-taught mathematician. But Grassmann’s work was origi-
nally ignored because he tried to construct a highly abstract self-contained theory, independent
of the rest of mathematics, containing nonstandard terminology and notation, and he had a
tendency to mix mathematics with obscure philosophy. Grassmann published a complete re-
vision of his work in 1862 but with no more success. Only later was it realized that he had
formulated the concepts we now refer to as linear dependence, bases, and dimension. The
Italian mathematician Giuseppe Peano (1858–1932) was one of the few people who noticed
Grassmann’s work, and in 1888 Peano published a condensed interpretation of it. In a small
chapter at the end, Peano gave an axiomatic definition of a vector space similar to the one
above, but this drew little attention outside of a small group in Italy. The current definition is
derived from the 1918 work of the German mathematician Hermann Weyl (1885–1955). Even
though Weyl’s definition is closer to Peano’s than to Grassmann’s, Weyl did not mention his
Italian predecessor, but he did acknowledge Grassmann’s “epoch making work.” Weyl’s success
with the idea was due in part to the fact that he thought of vector spaces in terms of geometry,
whereas Grassmann and Peano treated them as abstract algebraic structures. As we will see,
it’s the geometry that’s important.