## Linear independence |

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In the theory of **linearly dependent** if at least one of the vectors in the set can be defined as a **linearly independent**. These concepts are central to the definition of ^{[1]}

A vector space can be of

- definition
- geometric meaning
- evaluating linear independence
- natural basis vectors
- linear independence of basis functions
- space of linear dependencies
- see also
- references
- external links

A sequence of vectors from a *V* is said to be *linearly dependent*, if there exist scalars , not all zero, such that

where denotes the zero vector.

Notice that if not all of the scalars are zero, then at least one is non-zero, say , in which case this equation can be written in the form

Thus, is shown to be a linear combination of the remaining vectors.

A sequence of vectors is said to be *linearly independent* if the equation

can only be satisfied by for . This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence. In other words, a sequence of vectors is linearly independent if the only representation of as a linear combination of its vectors is the trivial representation in which all the scalars are zero.^{[2]} Even more concisely, a sequence of vectors is linear independent if and only if can be represented as a linear combination of its vectors in a unique way.

The alternate definition, that a sequence of vectors is linearly dependent if and only if some vector in that sequence can be written as a linear combination of the other vectors, is only useful when the sequence contains two or more vectors. When the sequence contains no vectors or only one vector, the original definition is used.

In order to allow the number of linearly independent vectors in a vector space to be *V* be a vector space over a *K*, and let {*v*_{i} | *i*∈*I*} be a *V*. The family is *linearly dependent* over *K* if there exists a *finite* family {*a*_{j} | *j* ∈ *J*} of elements of *K*, all non-zero, such that

A set *X* of elements of *V* is *linearly independent* if the corresponding family {*x*}_{x∈X} is linearly independent. Equivalently, a family is dependent if a member is in the closure of the

A set of vectors which is linearly independent and *x* over the reals has the (infinite) subset {1, *x*, *x*^{2}, ...} as a basis.