Homework Answers
We have proju1 (x) = [(x.u1)/(u1.u1)]u1 = [ (0-8-6+0)/(0+1+9+1)]u1 = -(14/11)(0,1,-3,-1)T = (0, -14/11,42/11,14/11)T, proju2 (x) = [(x.u2)/(u2.u2)]u2 = [(22-32+2+0)/(4+16+1+1)]u2 = -(4/11)(2,4,1,1)T = (-8/11,-16/11,-4/11,-4/11)T and proju3 (x) = [(x.u3)/(u3.u3)]u3 =[(11+0+2+0)/(1+0+1+9)]u3 = (13/11)(1,0,1,-3)T = (13/11,0,13/11,-39/11)T.
Now, let v = proju1 (x)+ proju2 (x)+ proju3 (x) = (0, -14/11,42/11,14/11)T+(-8/11,-16/11,-4/11,-4/11)T +(13/11,0,13/11,-39/11)T = (5/11, -30/11, 51/11, -29/11)T.
Also, let w = x-v = (11,-8,2,0)T- (5/11, -30/11, 51/11, -29/11)T= (116/11, -58/11, -29/11, 29/11)T.
Then x = v+w, where v, being a linear combination of u1,u2,u3 is in span{ u1,u2,u3 } and w = (29/11)(4,-2,-1,1)T is in span{u4}.
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