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Math Help - Showing that a set of linearly independent vectors are a basis?

  1. #1
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    Showing that a set of linearly independent vectors are a basis?

    I understand there are many ways and theorems to do this, but I have just been learning about the method where you find the coefficients of the basis using Gaussian elimination.

    I get how to do the manipulation, but I don't understand how it actually shows that a set of vectors span a space?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    For example, if we know \dim E=n and \{v_1,\ldots,v_n\}\subset E are linearly independent then, \{v_1,\ldots,v_n\} span E .
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  3. #3
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    If you are given that the set is independent, then the only thing remaining to prove is that they span the space. That is, you want to solve the equation a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n= v is solvable for any v in the space. If you are dealing with an n dimensional space, that reduces to n equations in n unknowns and, no matter what the right side is, that has a solution as long as Gaussian elimination does not give a row of all 0s.
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