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Homework Help: Lin Alg proof problem

  1. Oct 31, 2005 #1
    Problem: Explain why the columns of [itex]A^2[/itex] span [itex]\mathbb{R}^n[/itex] whenever the colums of A are linearly independent.

    By the theorem given in that section of the text, it is a logically equivalent fact that if the columns of [itex]A^2[/itex] are linearly independent, then they span [itex]\mathbb{R}^2[/itex] or
    [tex]\mathbb{R}^2=Span( \vec{a}_1 , \vec{a}_2 ) [/tex].

    How do I expand this definition from [itex]\mathbb{R}^2[/itex] to [itex]\mathbb{R}^n[/itex]?
    Last edited: Oct 31, 2005
  2. jcsd
  3. Oct 31, 2005 #2


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    If B is an nxn real matrix, then what can you say about whether or not its columns span Rn if its columns are linearly independent. Don't worry about B being a matrix. You know that it since it is nxn, it gives you n linearly independent columns, so you should know something about whether those columns span Rn. Once you know this, you should be able to say something about conditions on x if Bx = 0. You should also be able to say something about the invertibility of B. Can you get this far?
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