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{{#ev:youtube|https://www.youtube.com/watch?v=k7RM-ot2NWY|||||start=0}} | {{#ev:youtube|https://www.youtube.com/watch?v=k7RM-ot2NWY|||||start=0}} | ||
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?==== | |||
{{#ev:youtube|https://www.youtube.com/watch?v=k7RM-ot2NWY|250|left|||start= 495&end= 545&loop=1}} | |||
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. | |||
span | |||
basis of a vector | |||
linearly dependent | |||
linearly independent | |||
[[Category:Linear Algebra]] | [[Category:Linear Algebra]] |