Page 24 - Linear Algebra Done Right

Chapter 1. Vector Spaces
10
C is called a complex vector space. Usually the choice of F is either
obvious from the context or irrelevant, and thus we often assume that
F is lurking in the background without specifically mentioning it.
Elements of a vector space are called vectors or points. This geo-
metric language sometimes aids our intuition.
n
Not surprisingly, F is a vector space over F, as you should verify.
Of course, this example motivated our definition of vector space.
The simplest vector For another example, consider F , which is defined to be the set of
∞
space contains only all sequences of elements of F:
one point. In other
F ∞ ={(x 1 ,x 2 ,...) : x j ∈ F for j = 1, 2,... }.
words, {0} is a vector
space, though not a Addition and scalar multiplication on F ∞ are defined as expected:
very interesting one. (x 1 ,x 2 ,...) + (y 1 ,y 2 ,...) = (x 1 + y 1 ,x 2 + y 2 ,...),
a(x 1 ,x 2 ,...) = (ax 1 ,ax 2 ,...).
With these definitions, F becomes a vector space over F, as you should
∞
verify. The additive identity in this vector space is the sequence con-
sisting of all 0’s.
Our next example of a vector space involves polynomials. A function
p: F → F is called a polynomial with coefficients in F if there exist
a 0 ,...,a m ∈ F such that
2
p(z) = a 0 + a 1 z + a 2 z +· · ·+ a m z m
n
Though F is our for all z ∈ F. We define P(F) to be the set of all polynomials with
crucial example of a coefficients in F. Addition on P(F) is defined as you would expect: if
vector space, not all p, q ∈P(F), then p + q is the polynomial defined by
vector spaces consist
(p + q)(z) = p(z) + q(z)
of lists. For example,
the elements of P(F) for z ∈ F. For example, if p is the polynomial defined by p(z) = 2z+z 3
consist of functions on and q is the polynomial defined by q(z) = 7 + 4z, then p + q is the
3
F, not lists. In general, polynomial defined by (p + q)(z) = 7 + 6z + z . Scalar multiplication
a vector space is an on P(F) also has the obvious definition: if a ∈ F and p ∈P(F), then
abstract entity whose ap is the polynomial defined by
elements might be lists,
(ap)(z) = ap(z)
functions, or weird
objects. for z ∈ F. With these definitions of addition and scalar multiplication,
P(F) is a vector space, as you should verify. The additive identity in
this vector space is the polynomial all of whose coefficients equal 0.
Soon we will see further examples of vector spaces, but first we need
to develop some of the elementary properties of vector spaces.