Linear combinations, span, and basis vectors
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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.
span basis of a vector linearly dependent linearly independent