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February 20th, 2015, 05:39 PM  #1 
Newbie Joined: Feb 2015 From: United Kingdom Posts: 16 Thanks: 1  Equivalence proofs concerning matrices and systems of linear equations
Let $\mathbf A$ be the augmented $m × (n + 1)$ matrix of a system of $m$ linear equations with $n$ unknowns. Let $\mathbf B$ be the $m × n$ matrix obtained from $\mathbf A$ by removing the last column. Let $\mathbf C$ be the matrix in row reduced form obtained from $\mathbf A$ by elementary row operations. Prove that the following four statements are equivalent. (i) The linear equations have no solutions. (ii) If $c_1,\ldots, c_{n+1}$ are the columns of $\mathbf A$, then $c_{n+1}$ is not a linear combination of $c_1,\ldots, c_{n+1}$. (iii) $Rank(\mathbf A) \gt Rank(\mathbf B)$. (iv) The lowest nonzero row of $\mathbf C$ is $(0 0 · · · 0 0 1)$. My aim is to prove that (i)$\Leftrightarrow$(ii), (ii)$\Leftrightarrow$(iii) and (iii)$\Leftrightarrow$(iv). I am really struggling. Could I have guidance as to an effective way to approach these proofs? 
February 21st, 2015, 05:07 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
[QUOTE=CKKOY;223095]Let $\mathbf A$ be the augmented $m × (n + 1)$ matrix of a system of $m$ linear equations with $n$ unknowns. Let $\mathbf B$ be the $m × n$ matrix obtained from $\mathbf A$ by removing the last column. [quote] So the system of equations can be written as the matrix equation Bx= p where p is the 'last column' of the augmented matrix A. Quote:
Quote:
 
February 21st, 2015, 05:49 AM  #3 
Newbie Joined: Feb 2015 From: Warwick Posts: 2 Thanks: 0  How can this be applied to the question? I see what you have got from this and how, but I cant see how to use it at all? Sorry if Im being stupid :/

February 21st, 2015, 07:37 AM  #4 
Newbie Joined: Feb 2015 From: Warwick Posts: 2 Thanks: 0 
I am only struggling with i)<=>ii) and ii)<=>iii) now, please help 
February 22nd, 2015, 02:15 PM  #5 
Newbie Joined: Feb 2015 From: United Kingdom Posts: 16 Thanks: 1 
Are you still struggling with (i) <=> (ii) and (ii) <=> (iii)? Also, could you suggest a way to approach the proof that (iii) <=> (iv)? 
February 22nd, 2015, 03:16 PM  #6 
Newbie Joined: Feb 2015 From: United Kingdom Posts: 16 Thanks: 1 
I have managed to prove (iii) <=> (iv) now.


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equations, equivalence, linear, matrices, proofs, systems 
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