Basis (linear algebra)

In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space.

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More precisely, a subset B of a vector space V is a basis for V if it satisfies any of the following equivalent conditions:

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• B is both a set of linearly independent vectors and a generating set of V.
• B is a minimal generating set of V, i.e., it is a generating set but no proper subset of B is.
• B is a maximal set of linearly independent vectors, i.e., it is a linearly independent set but no proper superset of B is.
• Every vector in V can be expressed as a linear combination of vectors in B in a unique way.

Recall that a set B is called a generating set of V if every vector in V is a linear combination of vectors in B. This definition includes a finiteness condition: a linear combination is always a finite sum of the form a1v1 + ... + anvn.

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One can prove that every vector space has a basis. For spaces that cannot be finitely generated, Zorn's lemma is needed for the proof. All bases of a vector space have the same cardinality (number of elements), called the dimension of the vector space. The latter result is known as the dimension theorem.

Related Topics:
Zorn's lemma - Cardinality - Dimension - Dimension theorem

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