Let a_1 = [-6,-1,3,6,2,-3,-2,1],

a_2 = [-3,2,6,-3,-1,6,-1,2],

a_3 = [6,1,3,6,2,3,2,1],

a_4 [1,-6,-2,-1,3,2,-3,6]

Where each of the above are vectors (8x1)

1.) Show that the above vectors are orthogonal and then normalize them such that they all have length one. Call the new length-1 vectors u_1, u_2, u_3, u_4, respectively.

2.) Let U the 8x4 matrix which has the columns u_1, u_2, u_3, u_4. And, define:

y = [1,1,1,1,1,1,1,1], p = UU^(T)y, and z = y - p

Show that z is in-fact orthogonal to p.

My work:

#1 was easy; I showed the dot product between a_1 and a_2 is 0, between a_1 and a_3 is 0, etc etc.

I found that the length (norm) of all the vectors is 10; thus, in order to normalize them, I took the reciprocal and multiplied it by each of the vectors, to get u_1, u_2, u_3, u_4 as:

u_1 = [-3/5, -1/10, 3/10, 3/5, 1/5, -3/10, -1/5, 1/10]

u_2 = [-3/10, 1/5, 3/5, -3/10, -1/10, 3/5, -1/10, 1/5]

u_3 = [3/5, 1/10, 3/10, 3/5, 1/5, 3/10, 1/5, 1/10]

u_4 = [1/10, -3/5, -1/5, -1/10, 3/10, 1/5, -3/10, 3/5]

It's #2 that im stuck on.