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Concept
Linear independence
Summary
In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension.
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 from a vector space V is said to be linearly dependent, if there exist scalars not all zero, such that
where denotes the zero vector.
This implies that at least one of the scalars is nonzero, say , and the above equation is able to be written as
if and if
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 is said to be linearly independent if it is not linearly dependent, that is, 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. Even more concisely, a sequence of vectors is linearly independent if and only if can be represented as a linear combination of its vectors in a unique way.
If a sequence of vectors contains the same vector twice, it is necessarily dependent. The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is linearly independent if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is often useful.
Official source
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