# Check vectors form the basis

The *basis* in n-dimensional space is called the ordered system of n linearly independent vectors.

For the following description, intoduce some additional concepts.

Expression of the form:

λ_{1}
_{1} +
λ_{2}
_{2} + ... +
λ_{n}
_{n}

, where λ_{i} − some scalars and
i=[0; n] is called
*linear combination* of the vectors _{1} ,
_{2} , ... ,
_{n} .

If there are exist the numbers
λ_{1},
λ_{2}, ... ,
λ_{n} such as at least one of then is not equal to zero
(for example
λ_{j} ≠ 0)
and the condition:

λ_{1}
_{1} +
λ_{2}
_{2} + ... +
λ_{n}
_{n} =
0

is hold, the the system of vectors
_{1} ,
_{2} , ... ,
_{n} −
is called *linear dependent*.

If the equality above is hold if and only if, all the numbers
λ_{1} =
λ_{2} = ... =
λ_{n} =
0, then the system of vectors
_{1} ,
_{2} , ... ,
_{n} −
is called *linear independent*.

The *basis* can only be formed by the *linear independent* system of vectors.
The conception of linear dependence/independence of the system of vectors are closely related to the conception of
matrix rank.

Our online calculator is able to check whether the system of vectors forms the *basis* with step by step solution for free.