is called a vector space over the number field 
if to every pair
of elements of there corresponds a
sum 
, and
to every pair 
where 
and 
, there corresponds
an element 
, with
the properties 1-8: Vector Spaces
One of the fundamental concepts of linear algebra is the concept of vector space. At the same time it is one of the more often used concepts algebraic structure in modern mathematics. For example, many function sets studied in mathematical analysis are with respect to their algebraic properties vector spaces. In analysis the notion ``linear space'' is used unstead of the notion ``vector space''.
Definition A set is called a vector space over the number field if to every pair
of elements of there corresponds a
sum , and
to every pair where and , there corresponds
an element , with
the properties 1-8:
1. 
(commutability of
addition);
2. 
(associativity of
addition);
3. 
(existence of
null element);
4. 
(existence
of the inverse element);
5. 
(unitarism);
6. 
(associativity
with respect to number multiplication);
7. 
(distributivity
with respect to vector additism);
8. 
(distributivity
with respect to number additism).
The properties 1-8 are called the vector space axioms. Axioms 1-4 shows that is a commutative
group or an Abelian group with respect to vector addition. The second
correspondence is called multiplication of the vector by a number, and it
satisfies axioms 5-8. Elements of a vector space are
called vectors. If 
, then one speaks
of a real vector space, and if 
, then of a complex vector space. Instead of the notion ``vector
space'' we shall use the abbreviative ``space''.
Example Let us consider the set of all 
matrices with real
elements: 
The sum of two
matrices we define in usual way by the addition of the corresponding elements.
By multiplying the matrix by a real number 
we multiply all
elements of the matrix by this number. The simple check will show that
conditions 1-8 are satisfied. For example, let us check
conditions 3 and 4. We construct 
As 
the element
satisfied
condition 3 for arbitrary , and thus it is
the null element of the space . For the element
i.e., condition
4 is satisfied. Make sure of the valitidy of the remaining conditions 1-2 and
5-8.
The vector space in example is called n-dimensional real
arithmetical space or in short space . Declaring the
vector 
of
the space we
often use the transposed matrix 
In this
presentation we often use punctuation mark (comma, semicolon) to separate the
components of the vector, for example 
Example Let U, be a set
that consists of all pairs of real numbers 
We define addition
and multiplication by scalar in U as follows: 
Is the set
U a vector space?
Proposition Let be a vector space. For arbitrary vectors 
and number 
the following
assertions and equalities are valid:
of the vector
space is
unique;
of each is unique;
Become convinced of the trueness of these assertions! 
Example Let us consider the set of all 
matrices with
complex elements. The sum of this matrices will be defined by the addition of
the corresponding elements of the matrices. By multiplying the matrix by a
complex number one will multiply
by this number all the elements of the matrix. We leave the check that all conditions 1-8 are satisfied to the reader. This vector space
over the complex number field 
will be
denoted 
If we
confine ourselves to real matrices, then we shall get vector space 
over the number
field 
The
space 
will
be identified with the space 
and the space 
with the space
Example The set 
of all functions
is a
vector space (prove!) over the number field 
if 
and 