Let u equal v equal and let W be the subspace Let urightarro

Let u equal, v equal, and let W be the subspace

Let u^rightarrow = [1 9 0 3], v^rightarrow = [0 0 1 3], and let W the subspace of ropf^4 spanned by {u^rightarrow, v^rightarrow}. Find a basis for W^1.

Solution

Let w=(p,q,r,s)^T be an arbitrary element of W. Then u.w = 0 or, p+9q +3s =0 so that p = -9q-3s. Also, v.w = 0 so that r+3s = 0 or, r = -3s. Then w =(p,q,r,s)^T = ( -9q-3s, q,-3s,s)^T = q(-9,1,0,0)^T +s(-3,0,-3,1)^T. Then a basis for W is {(-9,1,0,0)^T , (-3,0,-3,1)^T }.