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:
λ1 1 + λ2 2 + ... + λn n
, where λi − some scalars and i=[0; n] is called linear combination of the vectors 1 , 2 , ... , n .
If there are exist the numbers λ1, λ2, ... , λn such as at least one of then is not equal to zero (for example λj ≠ 0) and the condition:
λ1 1 + λ2 2 + ... + λn 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 λ1 = λ2 = ... = λn = 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.
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