linearly independent

Let $V$ be a vector space over a field $F$ . We say that $v_1,\ldots, v_k\in V$ are linearly dependent if there exist scalars $\lambda_1,\ldots, \lambda_k\in F$ , not all zero, such that$$ \lambda_1 v_1+ \cdots +\lambda_k v_k = 0 .$$ If no such scalars exist, then we say that the vectors are linearly independent. More generally, we say that a (possibly infinite) subset $S\subset V$ is linearly independent if all finite subsets of $S$ are linearly independent.

In the case of two vectors, linear dependence means that one of the vectors is a scalar multiple of the other. As an alternate characterization of dependence, we also have the following.

Remark. Linear independence can be defined more generally for modules over rings: if $M$ is a (left) module over a ring $R$ . A subset $S$ of $M$ is linearly independent if whenever $r_1m_1+\cdots +r_nm_n=0$ for $r_i\in R$ and $m_i\in M$ , then $r_1=\cdots =r_n=0$ .