Linear combinations, span, and basis vectors: Difference between revisions
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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 | |||
[[Category:Linear Algebra]] | [[Category:Linear Algebra]] |
Revision as of 05:09, 7 November 2021
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