Find the coordinate vector of w relative to the basis S = (u1,u2) for R^2.

Question

Let u1=(3,-4), u2=(5,8), w=(1,1)

Answer 1

<1, 1> = a<3, - 4> + b<5, 8>;

We have linear system:

3a + 5b = 1, 12a + 20b = 4

-4a + 8b = 1, -12a + 24b = 3;

44b = 7; b = 7/44; 3a = 1 - 35/44; b = 3/44

w = <3/44, 7/44> in basis u₁ and u₂.