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Linear independence

Definition

A finite subset $\{v_1, ..., v_k\}$ of a vector space $V$ over a field $\mathbb{F}$ is linearly independent if the only option for $\alpha_1, ..., \alpha_k \in \mathbb{F}$ with $\sum_{i=1}^k \alpha_i v_i = 0$ is $\alpha_1 = ... = \alpha_k = 0$.

A subset $S \subset V$ is said to be linearly independent if every finite subset $T \subseteq S$ is linearly independent.

Properties

A linearly independent set $S$ is the basis for the subspace $\langle S \rangle \leqslant V$ which it spans.

Types

If a linearly independent set spans $V$, we say it is a basis for $V$.

Related theorems

all these pages adapted with probably insufficient credit from my university's lecture notes