Page 112 - A Course in Linear Algebra with Applications

96 Chapter Four: Introduction to Vector Spaces

(v) cd(v) = c(dv);
(vi) c(u + v) = cu + cv : (distributive law);
(vii) (c + d)v — cv + dv; (distributive law);
(viii) lv = v.

For economy of notation it is customary to use V to denote
the set of vectors, as well as the vector space. Since the vector
space axioms just listed hold for matrices, they are valid in
n
R ; they also hold in the other examples of vector spaces
described in 4.1.
More generally, we can define a vector space over an ar-
bitrary field of scalars F by simply replacing R by F in the
above axioms.
Certain simple properties of vector spaces follow easily
from the axioms. Since these are used constantly, it is as well
to establish them at this early stage.
Lemma 4.2.1
If u and v are vectors in a vector space, the following state-
ments are true:
(a) Ov = 0 and c 0 = 0 where c is a scalar;
(b) if u + v = 0, then u = —v;
(c) ( - l ) v = - v .

Proof
(a) In property (vii) above put c = 0 = d, to get Ov = Ov +
Ov. Add — (Ov) to both sides of this equation and use the
associative law (ii) to deduce that

0 = -(Ov) + Ov = (-(Ov) + Ov) + Ov,

which leads to 0 = Ov. Proceed similarly in the second part.
(b) Add —v to both sides of u + v = 0 and use the associative
law.
(c) Using (vii) and (viii), and also (a), we obtain

(
(
l
(
v + -l)v = v + -l)v = (1 + -l))v = Ov = 0.