Theorem: The determinant of the matrix of a linear transformation $T$ is the factor by which area/volume changes under $T$. If $T$ maps region $S$ to $T(S)$, then $\text{vol}(T(S)) = |\det(A)| \cdot \text{vol}(S)$. This is the basis for the Jacobian in multivariable change of variables.
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