= Solution
For a locally integrable $u$, the <change of variables formula> $y=Ax$ gives
$$
\int_{\mathbb R^n}u(Ax)\varphi(x)\,dx
=\frac1{|\det A|}
\int_{\mathbb R^n}u(y)\varphi(A^{-1}y)\,dy.
$$
This motivates
$$
\boxed{
\langle A^*u,\varphi\rangle
=\frac1{|\det A|}
\langle u,(A^{-1})^*\varphi\rangle}.
$$
Because pullback by an invertible linear map acts continuously on the <Schwartz space>, the right side is a continuous linear functional of $\varphi$. It therefore defines a <tempered distribution> and agrees with ordinary pullback when $u$ is a function.
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