Page 51 - Advanced Linear Algebra

Chapter 1

Vector Spaces

Vector Spaces

Let us begin with the definition of one of our principal objects of study.
Definition Let be a field, whose elements are referred to as scalars . A vector
-
space over is a nonempty set , whose elements are referred to as vectors,
=
-
together with two operations. The first operation, called addition and denoted
=
by b , assigns to each pair ² " Á # ³ of vectors in = a vector " b # in . The
second operation, called scalar multiplication and denoted by juxtaposition,
assigns to each pair ² Á "³  - d = a vector " in = . Furthermore, the
following properties must be satisfied:
1 )(Associativity of addition ) For all vectors "Á #Á $  = ,
" b ²# b $³ ~ ²" b #³ b $

2 )(Commutativity of addition ) For all vectors "Á #  = ,
"b# ~ #b"
3 )(Existence of a zero ) There is a vector = with the property that
b" ~ "b ~ "

for all vectors "= .
4 )(Existence of additive inverses ) For each vector "= , there is a vector
=
in , denoted by " c , with the property that
" b ²c"³ ~²c"³ b " ~