Let W be the subspace spanned by the given vectors Find a ba

Let W be the subspace spanned by the given vectors. Find a basis for W^perpendicular W_1 = [-6 -6 -18 -6], W_2 = [6 0 18 8], W_3 = [2 8 6 0]

Solution

v1=w1+3w3=[0 18 0 0]^T

v2=w1+w2=[0 -6 0 2]^T

w3=[2 8 6 0]^T

These three vectors span W as they are linearly indepdenent as w1,w2,w3

Let, v=[a b c d ] be in W\', W\' denotes W perpendicular

So, v.v1=0=18b

So, b=0

v.v2=0=-6b+2d=0

So, d=0

v.w3=0=2a+6c

a=-3c

v=[-3c 0 c 0]^T=c[-3 0 1 0]^T

Hence basis for W\' is {[-3 0 1 0]^T}