Find the coordinate vector of w relative to the basis S={u1,u2} for R^2

Question

u1= (12,-12)

u2= (4,4)

w= (72,0)

Answer 1

We must find real numbers a and b so that a<12, -12> + b<4, 4> = <72, 0>

So, 12a + 4b = 72 and -12a + 4b = 0.

From the second equation, a = (1/3)b.

Thus, 12(1/3)b + 4b = 72

8b = 72

So, b = 9 and a = 3

That is 3<12, -12> + 9<4, 4> = <72, 0>

The coordinate vector is <3, 9>.