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Columns of Matrix Form a Basis?

  1. Jan 23, 2012 #1
    Hi everyone,

    This not a homework question. I'm reviewing some linear algebra and I found this on a worksheet. I just need a hint on how to approach this problem.

    Let β=[itex]\{ v_1,v_2,...,v_n\}[/itex] be a basis for [itex]R^n [/itex]. Let M be the matrix whose columns are the basis vectors in β. Do the columns of [itex] M^2 [/itex] form a basis?

    I've played around with some examples and it seems to be true. I know that M has full rank and hence invertible, therefore, [itex] M^2 [/itex] is also invertible and has full rank. But this only means the columns of [itex] M^2 [/itex] spans the vector space. How do I show the columns are also linearly independent?
  2. jcsd
  3. Jan 24, 2012 #2
    If the columns are linearly dependent, then it doesn't have full rank. My definition of rank is the dimension of the space spanned by the columns.
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