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.

Check vectors form basis

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Volume of tetrahedron build on vectors online calculator