Linear Algebra Basics: True/False Questions and Rank Properties

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Homework Help Overview

The discussion revolves around true/false questions related to linear algebra concepts, specifically concerning the properties of subspaces and the rank of matrices.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of having a basis of 5 vectors in a subspace of R^n, questioning whether n must be greater than or equal to 5. There is also discussion regarding the rank of a 6x7 matrix and the number of linearly independent column vectors in its transpose.

Discussion Status

The discussion includes differing opinions on the first true/false question, with some participants asserting that the statement is true while others argue it is false. The second question about the rank of the matrix has prompted clarification on the definition of rank, with some guidance offered on the expected number of linearly independent vectors.

Contextual Notes

Participants are navigating homework questions that require careful consideration of definitions and properties in linear algebra, indicating a need for deeper understanding of the concepts involved.

Tony11235
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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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The first one is false, because it is not true for n greater than 5.

Claude.
 
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.
 
Okay, then it's false!
 
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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