$T: \Bbb R^3 \to \Bbb R^3$ and $S: \Bbb R^3 \to \Bbb R^4$ are matrix transformations whose standard matrices are $$T=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 3 & 4 \\ 1 & 5 & 0\end{bmatrix}$$
$$\text{and }[S] = \begin{bmatrix}3 & 1 & 2\\1 & 0 & 3\\0 & 2 & 2\\1 & 4 & -3\end{bmatrix}$$
Find the standard matrix of the transformation $S\cdot T: \Bbb R^3 \to \Bbb R^4$, and use that matrix to find $(S\cdot T) (1, 1,1)$
What I did was simple times $S$ to $T$ and found a new matrix is that considered the standard matrix?
and for the last part of the question do I times the standard matrix by $(1,1,1)$
Yes, that is what it says to do.
$S\cdot T$ is the matrix product of $S$ with $T$. The product of two transformation matrices is a transformation matrix.
$(S\cdot T)\vec u$ is the product of this product and the vector $\vec u$. It's the transformation of the vector.