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Suppose $U = [-P_1,P_1] \times [-P_2, P_2]$. Clearly this has volume(or area) $4P_1P_2$. Suppose I have a symmetric matrix $L = \left( \matrix{a \ b \\b \ c }\right)$. Is there a formula to calculate $L(U)$, the image of $U$ under $L$?

PS also if there is such a thing, is there a generalization of it to higher dimensions?

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    Do you want $L(U)$ or only its area? If the latter maybe use determinant. – 2017-01-03
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    See, e.g., "The Determinant: a Means to Calculate Volume," [PDF download](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALAPP/Peng.pdf). – 2017-01-03

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