Linear Algebra Questions

  • Thread starter Tony11235
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  • #1
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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.
 

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  • #2
Claude Bile
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The first one is false, because it is not true for n greater than 5.

Claude.
 
  • #3
HallsofIvy
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Tony11235 said:
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?
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.)

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