18 Chapter^1the reals. We recall that, in general, by a linear system (or vector space)
over a commutative field F = {a, (3, ... } is meant a set of elements X =
{x,y,z,. .. } together with an equal relation among its elements, a binary
operation in X called addition, and a multiplication of the elements of X
by those of F, satisfying the following conditions:
- x + y is uniquely defined and x + y E X.
- x + y = y + x.
- (x + y) + z = x + (y + z).
- There exists an element () E X such that x + () = x for all x E X.
- For every x E X there exists an element, denoted -x, such that
x+(-x) = B.
6. For every x EX and every a E F, the product ax is uniquely defined
and ax EX.
- a(f3x) = (af3)x.
- (a+,B)q; = ax+,Bx.
- a(x + y) = ax+ ay.
- μx = x, where μ is the unit of F.
It is clear that the complex number system C satisfies all of conditions 1
to 10, with F the commutative field of the reals.
The elements x 1 , x2, ... , Xn of X are said to be linearly independent
if the equation
aix1 + azXz + · · · + anXn = 0
where ak E F, implies that ak = 0 (k = 1, 2, ... , n). X is of dimension n
( n ~ 1) if there are in X n linearly independent elements x 1 , ... , x n and
if any element x E X can be expressed in the form
X = f31x1 + · · · + f3nXn
with /3k E F (k ~ 1, .. ., ii). The set {xi,xz,.. .,xn} is then called a
basis in X.The set {u1,u 2 }, where u 1 = (1,0) and u 2 =(0,1), is a basis in C, since
r1 u1 + r2u2 = (r1, 0) + (0, r2) = (r1, r2) = (0, 0)
implies that r 1 = r2 = 0, and we have already seen that for any complex
number (a, b) we have
(a, b) = au1 + bu2
Thus the complex number system is of dimension two over the reals.
However, C is of dimension one over itself.
An algebraic system A is said to be an algebra over the field F if A
is a ring and also a vector space over F, where the ring addition is the