# Linear Algebra Questions

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.

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

Claude.

HallsofIvy
Homework Helper
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?

I didn't come up with 5, it was a true/false/depends question on my homework.

HallsofIvy