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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 non-zero 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
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[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:
 Let $\mathbf C$ be the matrix in row reduced form obtained from $\mathbf A$ by elementary row operations.
So the matrix equation Bx= p is equivalent to the matrix equation Cx= q where q is the result of apply those row operations to p.

Quote:
 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 non-zero 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, 05:49 AM   #3
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Quote:
 Originally Posted by Country Boy 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. So the matrix equation Bx= p is equivalent to the matrix equation Cx= q where q is the result of apply those row operations to p.
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. Tags equations, equivalence, linear, matrices, proofs, systems Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post xsirmath Linear Algebra 1 November 3rd, 2014 09:36 AM kaspis245 Algebra 9 August 8th, 2014 10:07 AM yusufali Linear Algebra 3 September 19th, 2012 01:00 PM Jake_88 Linear Algebra 1 June 15th, 2010 05:59 AM jpcooper Algebra 2 November 13th, 2009 01:00 AM

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