Download 1.2. Vector Space of n-Tuples of Real Numbers

1.2. Vector Space of n-Tuples of Real Numbers
A point in an n-dimensional space can be denoted by an n-tuple of real numbers
A   a1,
, ai ,
, an 
where A is called an n-dimensional vector and the ai 's are its components.
contrast, a real number is called a scalar.
In
The set of all n-dimensional vectors is called the n-space or the vector space of
It is denoted by Rn.
n-tuples of real numbers.
Let A, B, C  Rn
and α, β be scalars.
Definitions
1. Equality:
AB 
ai  bi
2. Addition:
A+BC 
ai + bi  ci
3. Multiplication by scalar:
B  αA  Aα 
where i  1,…,n.
bi  α ai
Theorem 1.1
Vector addition is
1.
Commutative:
A B  B  A
2.
Associative:
A   B  C    A  B  C
3.
Distributive:
  A  B   A  B
Multiplication by scalar is
1.
Associative:
   A    A
2.
Distributive:
    A   A   A
Note that some operators on either sides of these equations do not have the same
meanings. For example, the + on the left of 2 denotes additions of real numbers
while the + on the right denotes vector addition.
Proof of the theorem is left as an exercise.
Miscellaneous
1.
The zero vector O   0,
AO  O  A  A
2.
The vector
 1 A   A
,0 is the identity element of addition, i.e.,
for all A.
is called the negative of A.
It is the inverse of A with
respect to addition, i.e.
A A  O
3. 0A  O and 1A  A .
These properties will be used in an abstract definition of a vector space or linear
algebra.