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Linear combinations, span, and basis vectors (view source)
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A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. | 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|1000|right|Let eet GO|frame||||start=0&end=100&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. | 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. | 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. | ||