Let U=span{x₁, x₂} where x₁=(1, 0, 1, 1) and x₂=(1, -1, 0, 1). Let x=(0, -1, 1, 1) in R⁴......please help!?

Question

a) compute p=proj∪(x).

b) Write x as the sum of a vector in U and a vector in U⊥.

c)Find the vector in U that is closest to x.

d) Find dim(U⊥)

I appreciate any help/insight that you can give. There is going to be another question like this on my next quiz and I have no idea how to do this.

Indica · Accepted Answer

(a) First generate an orthogonal basis for U, say {u₁,u₂}.

      u₁ = x₁ = (1,0,1,1), ||u₁||²=3
      
      u₂ = x₂ – (x₂•u₁)u₁/||u₁||² = (1,−1,0,1) – (2)(1,0,1,1)/3 = (⅓,-1,-⅔,⅓)
        For convenience write this as u₂ = (1,-3-2,1), ||u₂||²=15

       Projection of x is just sum of projections onto u₁ & u₂  :  (x•u₁)u₁/||u₁||²  + (x•u₂)u₂/||u₂||² 

        = (2)(1,0,1,1)/3 + (2)(1,-3,-2,1)/15 = (2/5)(2,-1,1,2)

        Rest are very easy.

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Let U=span{x₁, x₂} where x₁=(1, 0, 1, 1) and x₂=(1, -1, 0, 1). Let x=(0, -1, 1, 1) in R⁴......please help!?

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