I'm learning math and machine learning by myself, and I get confused when I see that some matrices are indexed by subscript whereas some are indexed by index. For example, the following Bellman equation is given by this video at around 39 minutes
$$\begin{align} \mathrm v &= R+\gamma P\mathrm v\\ \begin{bmatrix}v(1)\\v(2)\\\vdots\\ v(n)\end{bmatrix}&=\begin{bmatrix}R_1\\R_2\\\vdots\\ R_n\end{bmatrix}+\gamma \begin{bmatrix}P_{11} & P_{12} & \cdots &P_{1n}\\P_{21} & P_{22} & \cdots &P_{2n}\\\vdots & \vdots &\ddots & \vdots\\ P_{n1} & P_{n2} & \cdots &P_{nn}\end{bmatrix}\begin{bmatrix}v(1)\\v(2)\\\vdots\\ v(n)\end{bmatrix} \end{align}$$
why does it use $\mathrm v(1)$ rather than $\mathrm v_1$? And $R_1$ rather than $R(1)$? All I can see is that $R$ is capitalized whereas $\mathrm v$ is not, but I don't understand what on earth makes them different from each other. Please help me get this straight completely, thanks in advance.