Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.
For example, the function that corresponds to multiplication by the matrix: $$A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$ is $\quad f_A([x:y:z]):=[ax+by+cz:dx+ey+fz:gx+hy+iz]\quad$ where $[x:y:z]$ is a notation for an equivalence class of the vector with coordinates $(x,y,z).$