Let's say I want to sample from a high-dimensional distribution. Is it possible use a linear projection matrix to a lower-dimensional distribution, sample in the lower-dimensional space, and then transform the samples back into the original high-dimensional distribution?

For example, let's say I want to sample from a distribution in 4-space, such as P(x) = e^{-eta * x} where eta = [2, 3, 4, 2] and x = [x1, x2, x3, x4]^T. Let A be a 3x4 transformation matrix, and sample z = [z1, z2, z3]^T from the distribution Q(z) = e^{-eta * AT * z}. Then using our samples of z, can we somehow get generated samples of x from the original problem? This is just an example, but I was thinking more of in general.