Let S span s1 s2 R4 where s1 1 2 3 4T and s2 2 3 5 7T Det

Let S = span s1, s2 R4 where s1 = (1, 2, 3, 4)T and s2 = (2, 3, 5, 7)T. Determine the subspace W R4 which is perpendicular to S, i.e., all the vectors in W are perpendicular to all the vectors in S. Do this by finding a vector u which is perpendicular to both basis vectors of S. Continue until you can write W as span{ . . . }.

Solution

Let W contain vectors as

W = (w1, w2, w3, w4)

Since W is perpendicular to S1 and S2

W.S1 = W.S2 =0

i.e. w1+2w2+3w3+4w4 =0

2w1+3w2+5w3+7w4 =0

Only two equations in 4 variables so we can get dependent solutions

Eliminate w1 to have w2+w3+w4 =0

So W can have vectors as

(w1, w2, w3, -w2-w3)^T

W is span of (1 0 0 0)^t ( 0 1 0 0)^t ( 0 1 1 -2)^t