linearly independent

Let $V$ be a vector space over afield $F$. We say that $v_1,\mathrm{\dots },v_k\in V$ are linearly dependent if there exist scalars ${\lambda }_1,\mathrm{\dots },{\lambda }_k\in F$, not all zero, such that

$${\lambda }_1{v}_1+\mathrm{\cdots }+{\lambda }_k{v}_k=0.$$

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 thevectors is a scalar multiple of the other. As an alternatecharacterization of dependence, we also have the following.

Proposition 1.

Let $S\mathrm{\subset }V$ be a subset of a vector space. Then, $S$ islinearly dependent if and only if there exists a $v\mathrm{\in }S$ such that$v$ can be expressed as a linear combination of the vectors in theset $S\mathrm{\backslash \backslash }\mathrm{\{}v\mathrm{\}}$ (all the vectors in $S$ otherthan $v$ (http://planetmath.org/SetDifference)).

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 $r_1{m}_1+\mathrm{\cdots }+r_n{m}_n=0$ for $r_i\in R$ and $m_i\in M$, then $r_1=\mathrm{\cdots }=r_n=0$.