View source for 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.


====How do we know if a vector is linearly dependent?====
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Given a set of vectors, we can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0. If there are any non-zero solutions, then the vectors are linearly dependent. If the only solution is x = 0, then they are linearly independent.
In other words, if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be linearly independent.

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basis of a vector
linearly dependent
linearly independent


[[Category:Linear Algebra]]