Linear Combination - Linear Independence

Linear Independence

For some sets of vectors v₁,...,v_n, a single vector can be written in two different ways as a linear combination of them:

Equivalently, by subtracting these a non-trivial combination is zero:

If that is possible, then v₁,...,v_n are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors.

If S is linearly independent and the span of S equals V, then S is a basis for V.