Let u 2 1 and v 2 1 Show that h k is in span uv for all h

Let u = (2 -1), and v = (2 1). Show that (h k) is in span (u,v) for all h and k.

Solution

Let, (h,k)=au+bv

So,

h=2a+2b

k=-a+b

Adding 2 times second equation to first gives

h+2k=4b

b=(h+2k)/4

Subtracting 2 times second equation from first gives

h-2k=4a

b=(h-2k)/4

Hence for all (h,k) we found a,b so that

(h,k)=au+bv