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Linear Algebra Questions

  1. Jun 2, 2005 #1
    I have a few true/false/depends questions.

    If a subspace of R^n has a basis consisting of 5 vectors then n is greater than equal to 5. I say it's true because 5 linearly independent vectors span R^5. Is that correct?

    If the rank of a 6x7 matrix A is 3 then A^T has 5 linearly independent column vectors. I am not sure on this. Any help would be nice.
  2. jcsd
  3. Jun 2, 2005 #2

    Claude Bile

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    The first one is false, because it is not true for n greater than 5.

  4. Jun 3, 2005 #3


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    Yes, it is true. If the subspace has a basis consisting of 5 vectors, then it has dimension 5. Certainly it can't be a subspace of Rn if n is less than 5 (but Rn is a subspace of itself). Notice that you must say "greater than or equal to 5" because we are talking about a subspace, not Rn itself.
    (That's why Claude Bile's answer is incorrect. I suspect he confused the subspace with Rn itself.)

    No, if the rank is 3 then it has 3 linearly independent column vectors. How did you get 5?
  5. Jun 4, 2005 #4
    I didn't come up with 5, it was a true/false/depends question on my homework.
  6. Jun 4, 2005 #5


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    Okay, then it's false!
  7. Jun 4, 2005 #6


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    you need to go back and learn what rank means and what its properties are. otherwise even halls' (correct) statement will not do you much good.
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