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How does one multiply a list of matrices by a list of vectors, elementwise? For example, multiplying

A = {IdentityMatrix[2], 2*IdentityMatrix[2]}
x = {{1, 1}, {2, -2}}

should return

{{1, 1}, {4, -4}}

Neither Dot nor Times accomplishes this; both have the wrong dimensions. A cumbersome way would be

result = {{0, 0}, {0, 0}}
Do[result[[i]] = A[[i]].x[[i]], {i, 2}]

but surely there is a cleaner way.

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MapThread[Dot, {A, x}]   // TeXForm

$\left( \begin{array}{cc} 1 & 1 \\ 4 & -4 \\ \end{array} \right)$

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Perhaps just:

Dot @@@ Transpose[{A, x}]
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Following Belasarius in a somewhat more transparent fashion, you Apply[ ] the Dot function to A and x. The Transpose is used to get the A and x into a single list and the "1" causes the Apply to to work at the correct level.

 Apply[Dot, Transpose[{A, x}], 1]
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