Linear Dependence

linear-algebra

Definition

Linear Dependence

A set $M$ of vectors is called linearly dependent if it is possible to express a vectors from $M$ as a linear combination of other vectors in $M$. This means there exists a vectors $v\in M$ such that:

$$v\in [M\setminus \{v\}]$$

where $[M\setminus \{v\}]$ denotes the linear span of $M\setminus \{v\}$.

To test whether a set of vectors $v_1,v_2,\dots ,v_n$ is linearly dependent, one can check whether there exists a non-trivial linear combination of form:

$$\sum _{i=1}^n{\lambda }_i\cdot v_i=0$$

where ${\lambda }_i\in F$, $v_i\in V$ and $0$ is the zero vector of vector space $V$.