Linear Algebra Basics: True/False Questions and Rank Properties

In summary, the first statement is true because a subspace of R^n with a basis of 5 vectors must have dimension 5, meaning n must be greater than or equal to 5. The second statement is false because if a 6x7 matrix has a rank of 3, then it only has 3 linearly independent column vectors, not 5. It is important to understand the definition and properties of rank in order to answer such questions correctly.
  • #1
Tony11235
255
0
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
The first one is false, because it is not true for n greater than 5.

Claude.
 
  • #3
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
I didn't come up with 5, it was a true/false/depends question on my homework.
 
  • #5
Okay, then it's false!
 
  • #6
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.
 

1. What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It is an important tool in various fields such as physics, engineering, computer science, and economics. It helps in solving systems of equations, understanding geometric transformations, and analyzing data.

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