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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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    $y=ax+b$ is NOT a linear map unless $b=0$. And notice that the Jacobian matrix is the first-order derivatives in vector calculus.2012-07-25

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If the Jacobian is constant, your function is linear.