Linearly independent

In the theory of vector spaces, a set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent. These concepts are central to the definition of dimension.

A vector space can be of finite-dimension or infinite-dimension depending on the number of linearly independent basis 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 a basis for a vector space.

The vectors in a subset ${\displaystyle S=\{{\vec {v}}_{1},{\vec {v}}_{2},\dots ,{\vec {v}}_{n}\}}$ of a vector space V are said to be linearly dependent, if there exist a finite number of distinct vectors ${\displaystyle {\vec {v}}_{1},{\vec {v}}_{2},\dots ,{\vec {v}}_{k}}$ in ${\displaystyle S}$ and scalars ${\displaystyle a_{1},a_{2},\dots ,a_{k}}$ , not all zero, such that

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