Linear independence

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. A family $(v_1, \dots, v_k)$ of vectors of $V$ is linearly independent when the only linear combination of them that equals $0$ is the trivial one. This means that $$a_1 v_1 + \cdots + a_k v_k = 0 \implies a_1 = \cdots = a_k = 0$$ for all $a_1, \dots, a_k \in F$. An arbitrary family $(v_i)_{i \in I}$ is linearly independent when each finite subfamily is linearly independent. A family that is not linearly independent is linearly dependent.

Remarks

The vectors $(1, 0)$ and $(1, 1)$ in $\R^2$ are linearly independent; the vectors $(1, 2)$ and $(2, 4)$ are linearly dependent. A family that contains the zero vector, or the same vector twice, is linearly dependent. A family of at least two vectors is linearly dependent if and only if one of its vectors is a linear combination of the others. In $F^n$, a family of more than $n$ vectors is linearly dependent.

Explanations

L1 human

Vectors are linearly independent when none of them is redundant. No vector can be built from the others, so each one adds a new direction. Two arrows in the plane are independent exactly when they do not lie on one line.