Any help with this would be very appreciated.
Find the orthogonal projection of v on the null space of A if v=(-1,2,3,4) and
A= [1 -1 2 0]
[1 1 0 2]
I reduced to row echelon to solve the system but I'm not sure I am getting the nullspace right or if I need to obtain the least square solution as a similar example in the book shows. I need to make sure I am doing this properly. I would appreciate any help to solve correctly. Thank you.
Find the orthogonal projection of v on the null space of A if v=(-1,2,3,4) and
A= [1 -1 2 0]
[1 1 0 2]
I reduced to row echelon to solve the system but I'm not sure I am getting the nullspace right or if I need to obtain the least square solution as a similar example in the book shows. I need to make sure I am doing this properly. I would appreciate any help to solve correctly. Thank you.
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A = matrix whose columns are (1,−1,2,0) & (1,1,0,2)
You could find a basis for null(A) and then project onto that but why not project onto range space of A (for which you have a basis) and then subtract this from v.
You can project onto range(A) using standard result proj = Pv where P=A(A’A)⁻¹A’ is the orthogonal projection matrix. This is a least-squares approach.
This gives P = (1/3){ {1,0,1,1}, {0,1,-1,1}, {1,-1,2,0}, {1,1,0,2} } (by row)
Hence proj onto range space = Pv = (2,1,1,3)
Projection onto null-space = (−1,2,3,4) – (2,1,1,3) = (−3,1,2,1)
You could find a basis for null(A) and then project onto that but why not project onto range space of A (for which you have a basis) and then subtract this from v.
You can project onto range(A) using standard result proj = Pv where P=A(A’A)⁻¹A’ is the orthogonal projection matrix. This is a least-squares approach.
This gives P = (1/3){ {1,0,1,1}, {0,1,-1,1}, {1,-1,2,0}, {1,1,0,2} } (by row)
Hence proj onto range space = Pv = (2,1,1,3)
Projection onto null-space = (−1,2,3,4) – (2,1,1,3) = (−3,1,2,1)