LinearIndependence 2001-11-14 15:58:31 2008-05-26 11:56:43 Definition linearly dependent linear dependence \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \newtheorem{proposition}{Proposition} Let $V$ be a vector space over a field $F$. We say that $v_1,\ldots, v_k\in V$ are \emph{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 \emph{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. \begin{proposition} Let $S\subset V$ be a subset of a vector space. Then, $S$ is linearly dependent if and only if there exists a $v\in S$ such that $v$ can be expressed as a linear combination of the vectors in the set $S\backslash \{v\}$ (\PMlinkname{all the vectors in $S$ other than $v$}{SetDifference}). \end{proposition} \textbf{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 $r_1=\cdots =r_n=0$.