Basis (linear algebra)

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A set B of vectors in a vector space V is called a basis if every element of V
may be written in a unique way as a finite linear combination of elements of B.
The coefficients of this linear combination are referred to as components or
coordinates of the vector with respect to B.

The elements of a basis are called basis vectors.

Equivalently, a set B is a basis if its elements are linearly independent and
every element of V is a linear combination of elements of B.[1]
In other words, a basis is a linearly independent spanning set.

A vector space can have several bases;
however all the bases have the same number of elements,
called the dimension of the vector space.

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What is a basis?

Informally we say

A basis is a set of vectors that generates all elements of the vector space
and the vectors in the set are linearly independent.

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