# Linear Algebra Questions

1. Jun 2, 2005

### Tony11235

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. Jun 2, 2005

### Claude Bile

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

Claude.

3. Jun 3, 2005

### HallsofIvy

Staff Emeritus
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?

4. Jun 4, 2005

### Tony11235

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

5. Jun 4, 2005

### HallsofIvy

Staff Emeritus
Okay, then it's false!

6. Jun 4, 2005

### mathwonk

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.