Page 111 - A Course in Linear Algebra with Applications

4.2: Vector Spaces and Subspaces 95

Exercises 4.1
1. Give details of the geometrical interpretations of R 1 and
2
R .
2. Which of the following might qualify as vector spaces in
the sense of the examples of this section?
(a) the set of all real 3-column vectors that correspond to
line segments of length 1;
(b) the set of all real polynomials of degree at least 2;
(c) the set of all line segments in R 3 that are parallel to
a given plane;
(d) the set of all continuous functions of x defined in the
interval [0, 1] that vanish at x — 1/2.

4.2 Vector Spaces and Subspaces

It is now time to give a precise formulation of the defini-
tion of a vector space.

Definition of a vector space
A vector space V over R consists of a set of objects called
vectors, a rule for combining vectors called addition, and a
rule for multiplying a vector by a real number to give another
vector called scalar multiplication. If u and v are vectors, the
result of adding these vectors is written u + v, the sum of u
and v; also, if c is a real number, the result of multiplying v
by c, is written cv, the scalar multiple of v by c.
It is understood that the following conditions must be
satisfied for all vectors u, v, w and all real scalars c, d :

(i) u + v = v + u, (commutative law);
(ii) (u + v) + w = u + (v + w), (associative law);
(iii) there is a vector 0, called the zero vector, such that
v + 0 = v;
(iv) each vector v has a negative, that is, a vector —v
such that v + (—v) = 0;