1. consider the following subspaces of R^4
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 (A^TA)=Null(A)

Solution Summary

Topics included in this problem set include homogeneous linear equations, basis for a system, dimensions, normal equations, least squares solution, and projection matrices.

... 2/. Solution: The projection onto the subspace is the sum of the projections onto each ... 4/.Solution: Let A = (see attachment for matrix). The normal equation is. ...

... 10) The matrix inverse is. ... real world, when you need to make predictions for one ... but often still gives acceptable results using normal equations, a pseudoinverse ...

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... is a consistent asymptotically normal estimator of ... measured regressors simultaneous equations models dynamic ... a symmetric and idempotent projection matrix: 0 PZ ...

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... The regression equation is. ... want to calculate confidence intervals for your predictions you need ... is usually assumed that errors have a Normal distribution with ...

... The height of the tetrahedron is the projection of w on n: ... This is also the normal to plane M, therefore it must be ... So the equation of the plane M is simply: ...