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I
LINEAR ALGEBRA
A. Fields.
A field is a set of elements in which a pair of operations called
multiplication and addition is defined analogous to the operations of
multiplication and addition in the real number system (which is itself
an example of a field). In each field F there exist unique elements
called o and 1 which, under the operations of addition and multiplica-
tion, behave with respect to all the other elements of F exactly as
their correspondents in the real number system. In two respects, the
analogy is not complete: 1) multiplication is not assumed to be commu-
tative in every field, and 2) a field may have only a finite number
of elements.
More exactly, a field is a set of elements which, under the above
mentioned operation of addition, forms an additive abelian group and
for which the elements, exclusive of zero, form a multiplicative group
and, finally, in which the two group operations are connected by the
distributive law. Furthermore, the product of o and any element is de-
fined to be o.
If multiplication in the field is commutative, then the field is
called a commutative field.
B. Vector Spaces.
If V is an additive abelian group with elements A, B, .
..,
F a field with elements a,
b,.
.., and if for each a e F and A e V

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the product aA denotes an element of V, then V is called a (left)
vector space over F if the following assumptions hold:
1) a(A + B) = aA + aB
2) (a + b)A = aA + bA
3) a(bA) = (ab)A
4) 1A = A
The reader may readily verify that if V is a vector space over F, then
oA = O and aO = O where o is the zero element of F and O that of V.
For example, the first relation follows from the equations:
aA = (a + o)A = aA +oA
Sometimes products between elements of F and V are written in
the form Aa in which
case V is called a right vector space over F to
distinguish it from the previous case where multiplication by field ele-
ments is from the left. If, in the discussion, left and right vector
spaces do not occur simultaneously, we shall simply use the term
"vector
space."
C. Homogeneous Linear Equations.
If in a field F, ay, i = 1, 2,.
. . , m, j = 1,2,. . ., n are m • n ele-
ments, it is frequently necessary to know conditions guaranteeing the
existence of elements in F such that the following equations are satisfied:
a
il
X
l +
a
!2
X
2
(1)
The reader will recall that such equations are called linear
homogeneous equations, and a set of elements, x
x
, x
2
,. . . , x
n
of F, for which all the above equations are true, is called

a solution of the system. If not all of the elements
x
l
,
x
2
,. . . , x
n
are o the solution is called non-trivial; otherwise, it is called trivial.
THEOREM 1. A system of linear homogeneous equations always
has a non-trivial solution if the number of unknowns exceeds the num-
ber of equations.
The proof of this follows the method familiar to most high school
students, namely, successive elimination of unknowns. If no equations
in n > O variables are prescribed, then our unknowns are unrestricted
and we may set them all = 1.

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