What are the rank conditions for consistency of a linear algebraic system?

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Discussion Overview

The discussion revolves around the rank conditions for the consistency of a linear algebraic system. Participants explore definitions of rank, its application to both square and rectangular matrices, and the implications of these definitions on the consistency of systems of equations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant states that for a system of equations to be consistent, the rank of the coefficient matrix must equal the rank of the augmented matrix.
  • Another participant questions the definition of "rank" and proposes that it is the number of non-zero rows after row-reduction, asserting that this definition applies to non-square matrices as well.
  • A third participant cites their textbook's definition of rank as the dimension of the largest square submatrix with a non-zero determinant, expressing uncertainty about row reduction.
  • A later reply explains row reduction techniques, including swapping rows, multiplying a row by a constant, and adding a multiple of a row to another, but does not resolve the initial question about the rank condition.

Areas of Agreement / Disagreement

Participants express differing definitions and understandings of rank, leading to a lack of consensus on how to apply these definitions to the concept of consistency in linear systems.

Contextual Notes

There are unresolved assumptions regarding the definitions of rank and the applicability of row reduction techniques to different types of matrices. The discussion does not clarify how these definitions interact with the concept of consistency.

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what are the rank conditions for consistency of a linear algebraic system?
my proffessor said that the coefficient matrix augmented with the column value matrix must have the same rank as the coefficient matrix for consistency of the system of equations. however does the term rank apply to rectangular matrices such as an augmented matrix? I if so how is it calculated also I would like to know how to deduce this condition by reasoning [the derivation].
 
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Well, what is your definition of "rank"? The definition I would use is that the rank of a matrix is the number of non-zero rows left after you row-reduce the matrix. Obviously, that idea applies to non-square matrices. In fact, if you append a new column to a square matrix, to form the "augmented matrix", any non-zero row, after row-reduction, for the square matrix will still be non-zero for the augmented matrix- add values on the end can't destroy non-zero values already there. The only way the rank could be changed is if you have non-zero values in the new column on a row that is all zeroes except for that, so that the augmented rank has greater rank than the original matrix. That tells you that one of your matrices has reduced to 0x+ 0y+ 0z+ ...= a where a is non-zero and that is impossible. If there is no such case, you have at least one solution to each equation. Yes, the system is consistent if and only if the rank of the coefficient matrix is the same as the rank of the augmented matrix.
 
Well my textbook says that the rank of a matrix is the dimension of the largest square sub matrix whose determinant is non-zero. we just started the chapter so I'm not sure what you mean by row reduction, could you please elaborate? I'm sorry, I konw that these are really simple questions but I'd like to know the answers nevertheless... thanx.
 

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