Definition 1.3.1.
A set of vectors
of the vector space 
(over the field 
)
is said to be linearly dependent if
Definition 1.3.2.
A set of vectors of the space
(over the field )
is said to be linearly independent if it is not linearly dependent.
Example 1.3.1.* Let us check
if the set 
is linearly independent in the vector
space 
of all polynomials of at most degree n with real coefficients.
Let us consider the equality
It is well-known in algebra that a polynomial is identically null if all
its coefficients are zeros. Thus we get the system
This system has only a trivial solution. The set U is linearly
independent.
Problem 1.3.1.* Prove that each set of vectors that contains the null vector is linearly dependent.
Problem 1.3.2.* Prove that if the column-vectors of determinant are linearly dependent, then the determinant equals 0.
Definition 1.3.3. A
subset 
of the set 
of vectors of the vector space
is called a maximal linearly independent subset if V is linearly
independent and it is not a proper subset of any linearly
independent subset of the set U.
Proposition 1.3.1. If V is a maximal linearly
independent subset of the set U, then 
Proof. As 
by the definition of the span. To prove our
assertion, we have to show that 
Let, by antithesis, exist a vector 
of the subspace 
that does not belong to the subspace
Thus, the vector
cannot be expressed as a linear combination of vectors of V but
can be expressed as a linear combination
of vectors of 
when at least one vector 
is used, at which 
and 
is not expressable as a linear
combination of vectors of V. Set 
is linearly independent and contains the set V as a proper subset.
Hence V is not the maximal linearly
independent subset. We have got a contradiction to tha assumption.
Thus 
Q.E.D. 
Definition 1.3.4. A set 
of vectors of the vector space
is called a basis of the vector
space
if B is linearly independent
and each vector
of the space
can be expressed as a linear combination
of vectors of the set B, 
,
where coeffitients 
(i=1: n) are called coordinates of the vector
relative to the basis B.
Definition 1.3.5.
If the number of vectors in the basis B of the vector
space 
i.e., the number of elements of the set I, is finite, then this
number is called the dimension of the vector space
and denoted 
and the space
is called finite-dimensional or a finite-dimensional vector space.
If the number of vectors in the basis B of the vector
space
is infinite, then the vector space
is called infinite-dimensional or an infinite-dimensional vector
space.
Proposition 1.3.2. A subset B of the vectors
of the vector space
is a basis of the space iff it is the maximal linearly
independent subset.
Example 1.3.2. Vectors
form a basis in space 
Let us check the validity of the conditions in definition 1.3.4. As
the vector system 
is linearly independent, and, due to
arbitrary vector of the space 
can be expressed as a linear combination
of vectors 
Problem 1.3.3. Vector system
forms a basis in space 
Example 1.3.3. Vector system 
forms a basis in vector space 
of polynomials of at most degree 
Truely, the set
is linearly independent since
and each vector of space
(i.e., arbitrary polynomial of at most degree n) can be expressed
in the form
Definition 1.3.6. Two
vector spaces
and 
are called isomorphic, if there exist a one-to-one correspondence
between the spaces 
such that
Proposition 1.3.3. All vector
spaces (over the same number field )
of the same dimension are isomorphic.