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Math Help - Help: bases and scalars

  1. #1
    Junior Member
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    Help: bases and scalars

    I am having a lot of trouble wrapping my head around these vector spaces...

    How can I show that if {v1, v2, ... , vn} is a basis for a vector space V and c <> 0, then {cv1, v2, ... , vn} is also a basis for V?

    Any help would be appreciated, thanks!
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  2. #2
    Senior Member
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    If you know about matrices, you can easily construct a inversible matrix that transforms your basis into your second set of vectors, and so conclude that it's also a basis.
    But you can also show that each v_i , i=1,...,n , is a linear combination of {cv_1,...,v_n}, and then you've won because as {v_1,...v_n} is a basis, then your space has a dimension of n, so a subset with n vectors that spans your space is a basis.
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