- Let $X$ be a uniform random vector. Is any linear transformation $AX$ of $X$ still uniformly distributed? I know it is yes when $A$ is square and invertible, by using the change of variable formula. But not sure when $A$ is square and not invertible, or $A$ is not square.
- If $X$ and $Y$ are both uniform random variables (or vectors), will $(X^T,Y^T)^T$ be a random vector?