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Linear algebra help: Linear independence

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Let A be an m x n matrix of rank n. Suppose [tex]v_1, v_2, ..., v_k \in \mathbb{R}^n[/tex] and [tex]\{v_1, v_2, ..., v_k\}[/tex] is linearly independent. Prove that [tex]\{Av_1, Av_2, ..., Av_k\}[/tex] is likewise linearly independent.

    2. Relevant equations

    3. The attempt at a solution
    It says I need to use rank(A) = n, but I'm not sure how to use that info.
  2. jcsd
  3. Feb 22, 2012 #2


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    Rank n tells you that if you have a basis [itex]e_1, e_2, ..., e_n[/itex] of [itex]\mathbb{R}^n[/itex] then [itex]A(e_1), A(e_2), ..., A(e_n)[/itex] are linearly independent. I.e. A takes R^n into an n-dimensional subspace of R^m.
  4. Feb 23, 2012 #3


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    have you proven a subset of a linearly independent set is also linearly independent?

    if so, extend the v's to a basis.
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