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  1. vars = {x, y, z};
  2. m = {
  3.   {1, 3, 0},
  4.   {-1, 0, 1}
  5. };
  6. v == m.vars (* True *)
  7.    
  8. M = Table[Mij[i, j], {i, 1, 2}, {j, 1, 3}];
  9. Solve[M.{x, y, z} == {x + 3 y, z - x}, Flatten@M]
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