Linear Dependency Theorem for Vectors Relative to a Basis

Definition

In a vector space V over the field K with a basis B={v₁,...,v_n}, every vector v in V is linearly dependent on the vectors of the basis B.

Proof

If B is a basis of the vector space V and v is a vector in V, then there exist scalars such that

$$ v = a_1 v_1 + ... + a_n v_n $$

This is equivalent to stating

$$ v - a_1 v_1 - ... - a_n v_n = 0 $$

Thus, it is possible to derive the null vector { 0_v }

Since the scalar coefficient multiplying v equals 1, the linear combination is non-trivial.

$$ (1) v - a_1 v_1 - ... - a_n v_n = 0 $$

Hence, the set of vectors { v, v₁, ... , v_n } consists of linearly dependent vectors.

Concluding, we can state that

Adding a vector to a vector basis always results in a set of linearly dependent vectors.