Abstract Linear Algebra Notes
9/26
Definition – Let F be a set with at least two elements. Then
is a
field
if:
1.
F is an abelian group under + with additive identity element 0;
2.
F\{0} is an abelian group under
;
∙ with multiplicative identity 1
3.
Distribution laws hold:
Definition – G is a
group under
∗
if:
1.
G is closed under
∗
:
2.
Has identity element, e
3.
Associative laws hold
4.
Inverses:
Examples:
R – field of real numbers (under usual addition and multiplication)
C – field of complex numbers (“
“
“
“
“)
Q – field of rational numbers (“
“
“
“
“)
F
p
– field of integers modulo p where p is a prime
Definition 
characteristic of F “char(F)”
is the least positive integer n s.t.
If such n does not exist, then the characteristic of F is 0.
Notice:
char(R)=char(C)=char(Q)=0, char(F
p
)=p
Proposition – The characteristic of a field can’t be composite.
Pf:
Suppose that the characteristic of a field is a composite number n. So,
n=a
∙
b, where 1 < a,b < n and a,b
∈
F. Notice,
→

←
, since a < n, b < n, and n is the characteristic of the field. Therefore, n must
be prime
QED
Definition – A
vector space (linear space) over a field F
consists of the
following:
1.
A field F of scalars;
2.
A set V of objects (V
≠
∅
), called vectors;
3.
A rule (or operation), called vector addition, +: V x V
→
,
V where V is
;
an abelian group under addition
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A rule (or operation), called scalar multiplication,
*
: F x V
→
V, where:
(a)
(identity)
(b)
(associative law)
(c)
(dist. law).
Examples of vector spaces over a field F (more info view Hoffman pgs 2931):
1.
The ntuple space, F
n
2.
The space of
matrices,
3.
The space of functions from a space to a field
4.
The space of polynomial functions over a field F
5.
Let V be a vector space over F and let S be a nonempty set.
Let
= set of all
functions S to V.
Propositions – Let V be a vector space over F.
(1)
Then the additive identity, 0, is unique.
Pf:
Suppose there is two 0
1
and 0
2
. Then,
by using the definition of additive identity we have shown that the additive identity
is unique.
QED
(2)
If there exists a pair
s.t. , the w is the additive identity.
Pf:
Take any
and consider
Therefore, w is the additive identity element.
QED
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 Spring '08
 Staff
 Linear Algebra, Algebra, Vector Space, linear transformation, PF

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