vars = {x, y, z};
m = {
  {1, 3, 0},
  {-1, 0, 1}
};
v == m.vars (* True *)

M = Table[Mij[i, j], {i, 1, 2}, {j, 1, 3}];
Solve[M.{x, y, z} == {x + 3 y, z - x}, Flatten@M]