Given a random vector $X \in \mathbb{R}^k$, with a known pdf given by $f_X$. If $Y, Z \in \mathbb{R}^k$ are defined by $Y = AX$, $Z = BX$, where $A,B \in \mathbb{R}^{k\times k}$ are different, given, real-valued matrices.
I know how to calculate pdfs of $Y$ and $Z$ on their own. But how do I derive the joint pdf of $Y$ and $Z$?
If it helps to be more specific, $f_X$ is a mixture of $0$-mean multivariate gaussians, each component in the mixture with a different, diagonal covariance matrix (but not of the form $\Sigma = \sigma^2 I$).
Any help would be much appreciated.
For some context:
My goal is to check for the $f_X$ mentioned above, and a specific $A$ and $B$, whether the vectors $Y$ and $Z$ are independent. This means I need to check whether the joint distribution of $Y$ and $Z$ factorises into the product of the marginals. There are at least some cases when this is true: if, for example, $X \sim \mathcal{N}(0,\sigma^2 I)$ and $A$ and $B$ are projections onto orthogonal subspaces. But proving it is not true in my case would also be helpful. Hence my need to derive the joint distribution of $Y$ and $Z$.