What is the relationship between the rank of a matrix and its transpose?

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Discussion Overview

The discussion centers on the relationship between the rank of a matrix and the rank of its transpose, specifically exploring whether Rank (A^T) equals Rank (A) for an mxn matrix A. Participants are considering theoretical aspects and proofs related to this concept.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • John proposes that Rank (A^T) = Rank (A) may relate to the equivalence of Row Rank and Column Rank.
  • Another participant questions the definitions of column rank of A and row rank of A^T, seeking clarification on their relationship.
  • One participant asserts that the column rank of A is equal to the row rank of A^T, suggesting that they are equal and thus Rank (A^T) = Rank (A).
  • A suggestion is made to use row reduction or general theory of dimension to support the argument.

Areas of Agreement / Disagreement

There is some agreement that the ranks are related, but the discussion includes differing levels of certainty and approaches to proving the relationship. No consensus is reached on a definitive proof or explanation.

Contextual Notes

Participants have not fully defined their terms, and there may be assumptions regarding the properties of matrix ranks that are not explicitly stated. The discussion does not resolve the mathematical steps necessary for a proof.

Who May Find This Useful

Individuals interested in linear algebra, particularly those studying matrix theory and properties of matrix ranks.

oldmathguy
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Hi, I'm new to the forum but have watched it for some time. I am trying to prove that Rank (A^T) = Rank (A) with A being mxn matrix. I suspect that it has to do with Rank (A) = Row Rank (A) = Column Rank (A) -and- A^T simply being rows / columns transposed but am unsure how to prove. Thanks, John.
 
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What's the column rank of A? And what's the row rank of A^t?

What's the relation between these two?
 
They're equal !

Col Rank (A) = Row Rank (A^T) so dim (A) = Col Rank (A) = Row Rank (A^T) = Rank (A^T). Thanks ! John.
 
use row reduction. or the general theory of dimension.
 

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