
Let u=[3,4] and v=[1,1]. Find scalars s and t so that
the equation,

Enter the vectors, 
> u := [3,4]: v := [1,1]:
and write down the equation as, 
> Eq := evalm(s*[0,3]+t*u) = v;
Eq := [3 t, 3 s + 4 t] = [1, 1]
in order for these two vectors to be equal they must have their coordinates the same. We therefore obtain a system of two equations with two unknows. 
> Eq1 := lhs(Eq)[1] = rhs(Eq)[1];
Eq1 := 3 t = 1> Eq2 := lhs(Eq)[2] = rhs(Eq)[2];
Eq2 := 3 s + 4 t = 1
and the solution is: 
> solut := solve({Eq1,Eq2},{s,t});
solut := {s = 1/9, t = 1/3}

In this simple case we could have just obtained the answer by simple inspection but the fancy way of grabbing the coordinates of the "lefthandsideofEq" with lhs(Eq)[1] can still be used in more complicated problems with the same amount of writing! 