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February 20th, 2015, 05:39 PM   #1
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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?
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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?
Thanks from CKKOY
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February 21st, 2015, 05:49 AM   #3
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Quote:
Originally Posted by Country Boy View Post

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