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True or False?

Let $A$ be an $m \times n$ matrix. If $m > n$ , then the linear system $Ax=b$ is inconsistent for at least one vector $b$ in $\mathbb{R}^n$

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It should be $b\in \mathbb{R}^m$ –  Dominic Michaelis Feb 14 '13 at 16:39
    
^^^ why is this? –  Johnathon Svenkat Feb 14 '13 at 17:01
    
because if $A\in\mathbb{R}^{m \times n}$ and $x\in \mathbb{R}^n$ than $A\cdot x \in \mathbb{R}^m$ –  Dominic Michaelis Feb 14 '13 at 17:04
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1 Answer 1

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The set of all $b$ for which $Ax=b$ has a solution is the image of $A$. Which is the maximum dimension of the image of an $m\times n$ matrix?

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The image is $\{b\in \mathbb{R}^m : \exists x \in \mathbb{R}^n : Ax=b\}$ –  Dominic Michaelis Feb 14 '13 at 16:39
    
...or, which is the same, $\{Ax \colon x\in \mathbb R^n\}$ –  Emanuele Paolini Feb 14 '13 at 16:43
    
sorry, I still don't understand what you mean by 'image' ? what exactly is that referring to? –  Johnathon Svenkat Feb 14 '13 at 16:45
    
For linear functions $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ we know that dim(kern($f$))+dim(image($f$))=$n$ –  Dominic Michaelis Feb 14 '13 at 16:51
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