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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.

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