Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\mathbb{R}^n$

share|cite|improve this question
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
up vote 0 down vote accepted

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?

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.