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When determining whether or not a map/transformation is linear or non-linear, how can the Jacobian matrix be used? A linear equation in two variables is one that may be written in the form y = ax + b, but how do know if it is non-linear?

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You can check if the transform is linear. If $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ satisfies $T(ax+by)=aT(x)+bT(y)\ \forall\ x,y\in\mathbb{R}^n$ and $a,b\in\mathbb{R}$, then the transform $T$ is linear. Otherwise, it is non-linear. –  Daryl Jul 25 '12 at 0:39
    
Here I assumed that you are mapping from one vector space to another, as you didn't specify this information. –  Daryl Jul 25 '12 at 0:41
    
$y=ax+b$ is NOT a linear map unless $b=0$. And notice that the Jacobian matrix is the first-order derivatives in vector calculus. –  chaohuang Jul 25 '12 at 1:02
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up vote 4 down vote accepted

If the Jacobian is constant, your function is linear.

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