Linear Algebra (Meaning of Rank)

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Homework Help Overview

The discussion revolves around the concept of rank in linear algebra, specifically regarding the relationship between the rank of an n x n matrix and the number of linearly independent row vectors in that matrix.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore whether the statement about the rank of an n x n matrix holds true universally or if there are exceptions for matrices of different dimensions. Questions arise about the implications of having more rows than columns and the definitions of row rank and column rank.

Discussion Status

There is an active exploration of the definitions and implications of rank in different matrix configurations. Some participants express uncertainty about the limitations of the statement when applied to non-square matrices, while others reference external sources to support their points. The conversation reflects a mix of agreement and differing interpretations without reaching a consensus.

Contextual Notes

Participants note the specific focus on n x n matrices and question the reasoning behind this limitation, suggesting it may be a trick in the question. There is also mention of external resources, such as a Wikipedia article, to clarify concepts related to rank.

DanielFaraday
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Homework Statement



True or False:
If A is an n x n matrix, then the rank of A equals the number of linearly independent row vectors in A.

Homework Equations



None


The Attempt at a Solution



Okay, I know this is a ridiculously easy question, but I'm wondering if there is a catch to it. My inclination is to say this is true. But wouldn't this be true for ANY matrix? Why does the question limit it to an n x n matrix? Are they just being tricky?
 
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DanielFaraday said:

Homework Statement



True or False:
If A is an n x n matrix, then the rank of A equals the number of linearly independent row vectors in A.

Homework Equations



None


The Attempt at a Solution



Okay, I know this is a ridiculously easy question, but I'm wondering if there is a catch to it. My inclination is to say this is true. But wouldn't this be true for ANY matrix? Why does the question limit it to an n x n matrix? Are they just being tricky?

Well, if it were n x m with n not equal to m, then the row rank and column rank would not generally be the same, right?

Also, if you have more rows than columns, what can you say about the linear independence of the rows...?

http://en.wikipedia.org/wiki/Rank_(linear_algebra )

.
 
Last edited by a moderator:
berkeman said:
Well, if it were n x m with n not equal to m, then the row rank and column rank would not generally be the same, right?

Also, if you have more rows than columns, what can you say about the linear independence of the rows...?

http://en.wikipedia.org/wiki/Rank_(linear_algebra )

.

Thanks for your reply. In the wikipedia article that you cited (which I had read before posting this question) it says in the third sentence: "Since the column rank and the row rank are always equal, they are simply called the rank of A". I believe this statement is correct, which disagrees with your first point. Am I missing something?
 
Last edited by a moderator:
DanielFaraday said:
Thanks for your reply. In the wikipedia article that you cited (which I had read before posting this question) it says in the third sentence: "Since the column rank and the row rank are always equal, they are simply called the rank of A". I believe this statement is correct, which disagrees with your first point. Am I missing something?

No, that sentence was my point. if n=m, then you use the term "rank" instead of differentiating between rows and columns. I think your original answer is correct, just that it cannot be extended always to cases where n is not equal to m. Hope I'm not just adding confusion here...
 
Okay, that makes perfect sense. Thank you!
 

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