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.

Proposition 1.

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 $S\backslash\{v\}$ (all the vectors in $S$ other than $v$ (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 $r_{1}m_{1}+\cdots+r_{n}m_{n}=0$ for $r_{i}\in R$ and $m_{i}\in M$, then $r_{1}=\cdots=r_{n}=0$.

Title linearly independent LinearlyIndependent 2013-03-22 11:58:40 2013-03-22 11:58:40 rmilson (146) rmilson (146) 30 rmilson (146) Definition msc 15A03 linear independence linearly dependent linear dependence