A neat way of representing a matrix, where each column is a different matrix-vector product?

Question

The background of my problem is quite complex, so I won't go through it here. It basically boils down to this:
Is there any neat way (matrix product etc.) to represent the $t\times r$ matrix \begin{equation} X= \begin{bmatrix} & \\ A_1\beta_1 & ... & A_r\beta_r\\ & \end{bmatrix} \end{equation} where each $A_i \in \mathbb{R}^{t\times n}$ and $\beta_i \in \mathbb{R}^n$? An optimal solution would involve all of the $\beta$-vectors in a single matrix.