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Math Help - Matrix proof involving invertibility and bases

  1. #1
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    Matrix proof involving invertibility and bases

    Ok, so I need to do the following:

    Let A be an invertible n by n matrix and let {v1, ..., vn} be a basis for R^n. Prove that {Av1, ..., Avn} is also a basis for R^n.

    Any ideas on how I could go about doing this??
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  2. #2
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    Quote Originally Posted by crymorenoobs View Post
    Ok, so I need to do the following:

    Let A be an invertible n by n matrix and let {v1, ..., vn} be a basis for R^n. Prove that {Av1, ..., Avn} is also a basis for R^n.

    Any ideas on how I could go about doing this??

    You only need to show that \{Av_1,\ldots,Av_n\} is lin. indep.:

    For a_i\in\mathbb{F}= the definition field , 0=\sum^n_{i=1}a_iAv_i=\sum^n_{i=1}A(a_iv_i)=A\left  (\sum^n_{i=1}a_iv_i\right) . Now, since A is invertible we have that \ker(A)=Null(A)=\{0\} , and then use that \{v_1,\ldots,v_\} is a basis ...

    Tonio
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  3. #3
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    Oh ok, thanks so much!
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