Given a system of $100$ different states $\hat p_i$ and a function that assigns a value to each of the states at a given time, such that: $\overrightarrow m (t)= \{m_1(t),...,m_{100}(t)\}$. I make the assumption that the values evolve in the following way: $\overrightarrow m(t+1) = B . \overrightarrow m(t)$ where B is a random $5\times 5$ matrix.
I want to show that:
$\overrightarrow m (t) = B^T . \overrightarrow m (0)$ where $B^T$ is the transpose of the random $5\times 5$ matrix.
I'm not sure how to approach this question. Shall I express the vector in terms of the states and then rearrange? Furthermore, shall I set the matrix $B$ as a $100\times 100$ (number of states) rather than a $5\times 5$?
Thanks!