Linear independence

In linear algebra, a set of vectors is said to be linearly independent if there exists no vector in the set that is equal to a linear combination of the other vectors in the set. If such a vector exists, then the vectors are said to be linearly dependent. Linear independence is part of the definition of linear basis.

A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

A sequence of vectors ${\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{k}}$ from a vector space V is said to be linearly dependent, if there exist scalars ${\displaystyle a_{1},a_{2},\dots ,a_{k},}$ not all zero, such that

where ${\displaystyle \mathbf {0} }$ denotes the zero vector.

If ⁠${\displaystyle k=1}$⁠, this implies that a single vector is linear dependent if and only if it is the zero vector.

If ⁠${\displaystyle k>1}$⁠, this implies that at least one of the scalars is nonzero, say ${\displaystyle a_{1}\neq 0}$, and the above equation is able to be written as

Thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others.

A sequence of vectors ${\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}}$ is said to be linearly independent if it is not linearly dependent, that is, if the equation