Questions:
Normal Equations and Projection Matrices
1. Consider the following subspaces of R4
V=span{v1,v2,v3}, W=span{w1,w2,w3}
where v1=(1,2,1,-2)T w1=(1,1,1,1)T
v2=(2,3,1,0)T w2=(1,0,1,-1)T
v3=(1,2,2,-3)T w3=(1,3,0,-4)T
a)Find two systems of homogeneous linear equations whose solution spaces are V and W, respectively.
b)Find a basis for V∩W by joining the systems obtained in (a) in one system and solving it. Explain why the solution of the latter is exactly V∩W.
c) Find dim V and dim W and calculate the dimension of V+W using the dimension formula
2. Solving the normal equation, find the least squares solution x0 of the system of linear equations
x+y+2z=2
x+2y+3z=-3
x+2y+z=4
x+y+4z=1
should use Maple or Matlab for calculations
3. Let W=span {u,v,w} where
u=(1,1,1,1)T
v=(1,1,1,0)T
w=(0,1,1,1)T and b=(2,1,0,1)T
a) Find the projection w=projw(b) of b onto W
b) Find the projection matrix P onto W and check that w=Pb
c) Find w'∈W⊥ such that b=w+w' 4. Prove that Null (ATA)=Null(A)