Using Least Squares to find Orthogonal Projection

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Homework Statement

Use least squares to find the orthogonal projection of u onto the subspace of R4 spanned by the vectors v1, v2, and v3.

Relevant Equations

u = (0,5,4,0) v1 = (6,0,0,1) v2 = (0,1,-1,0) v3 = (1,1,0,-6)

I'm a little confused how to do this homework problem, I can't seem to obtain the correct answer. I took my vectors v1, v2, and v3 and set up a matrix. So I made my matrix:

V = [ (6,0,0,1)^T, (0,1,-1,0)^T, (1,1,0,-6)^T ] and then I had
u = [ (0,5,4,0) ^T ].

I then went to solve using least squares. So I ended up doing V^TV = V^Tu.

I took that, obtained my augmented matrix:

[ (37,0,0)^T, (0,2,1)^T, (0,1,38)^T, (0,1,5)^T ].

I solved it and everything worked fine, I got the answers: x₁ = 0, x₂ = 11/25, x₃ = 3/25.

I noticed to that I could use QR factorization of V then solve Q^Tu and the least squares solution would be x' satisfies Rx = Q^Tu, we can then obtain our answer, but again this solution only has three components, why is the solution asking for four?

However, the solution requires four components. I am unsure of where I am going wrong with this problem, any help would be appreciated.

 

[ehild]

Homework Statement:: Use least squares to find the orthogonal projection of u onto the subspace of R4 spanned by the vectors v1, v2, and v3.
Relevant Equations:: u = (0,5,4,0) v1 = (6,0,0,1) v2 = (0,1,-1,0) v3 = (1,1,0,-6)

I'm a little confused how to do this homework problem, I can't seem to obtain the correct answer. I took my vectors v1, v2, and v3 and set up a matrix. So I made my matrix:

V = [ (6,0,0,1)^T, (0,1,-1,0)^T, (1,1,0,-6)^T ] and then I had
u = [ (0,5,4,0) ^T ].

I then went to solve using least squares. So I ended up doing V^TV = V^Tu.

I took that, obtained my augmented matrix:

[ (37,0,0)^T, (0,2,1)^T, (0,1,38)^T, (0,1,5)^T ].

I solved it and everything worked fine, I got the answers: x₁ = 0, x₂ = 11/25, x₃ = 3/25.

I noticed to that I could use QR factorization of V then solve Q^Tu and the least squares solution would be x' satisfies Rx = Q^Tu, we can then obtain our answer, but again this solution only has three components, why is the solution asking for four?

However, the solution requires four components. I am unsure of where I am going wrong with this problem, any help would be appreciated.

You have to find the orthogonal projection of u onto the subspace of R4 spanned by the vectors v1, v2, and v3. x1, x2, x3 you got are the componets of this projection vector P with respect to the basis v1, v2, v3, that is
P= x1 v1 +x2 v2+x3 v3 , a linear combination of three four-dimensional vectors .