Linear combinations, span, and basis vectors: Difference between revisions

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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?====
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====How do we know if a vector is linearly dependent?====
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.

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