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True or False?

Let $A$ be an $m x n$ matrix. If $m \lt n$, then for each $b$ in $\mathbb R^{m}$ the linear system $Ax=b$ is either inconsistent or has infinitely many solutions.

Can anyone explain this only using beginner level linear algebra material (e.g. vector spaces, bases, linear independence/dependence) ?

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1 Answer 1

up vote 1 down vote accepted

Maybe this is how it works for you. Because of $m <n$ the kernel of $A$ can not be trivial. Because it's not trivial there is $v\in \text{kern}(A)$ with $v\neq 0$. Now let $x$ be a solution of $$A x=b.$$ We can construct infinitely many others solutions from this one, taking $$ A(x+\gamma v)=Ax + A (\gamma v)=Ax+ \gamma A v =Ax=b$$ for any $\gamma \in \mathbb{R}$.

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by kernel, do you mean null space? –  Johnathon Svenkat Feb 14 '13 at 17:22
    
yeah sry english is a foreign language :( –  Dominic Michaelis Feb 14 '13 at 17:23
    
no worries. thanks for your help! –  Johnathon Svenkat Feb 14 '13 at 17:25

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