# Linear Algebra w/ matrix + linear system question

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 $\mathbb{R}^n$

-
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

## 1 Answer

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?

-
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