1
Examples of projection/least square problems
Problem 1. Let S=R(A), where
A=
−1−4
−1−3
2−3
and b=
−1
2
2
.
Find the projection pand residual rof bon S.
Solution. We have that
ATA=6 1
1 34and ATb=3
−8.
Solving the linear system ATAc =ATbin cwe obtain c= [110/203,−51/203]T. Hence
p=Ac = [94/203,43/203,373/203]T
and
r=b−p= [−297/203,363/203,33/203]T.
Problem 2. Find the least square linear fit to the data
x−1 0 1 2
y−2−1 0 3 .
Solution. Let
A=
1−1
1 0
1 1
1 2
and b=
−2
−1
0
3
.
We have that
ATA=4 2
2 6and ATb=0
8.
Solving the linear system ATAc =ATbin cwe obtain c= [−4/5,8/5]T. Hence
y= (−4/5)x+ 8/5
is the least square linear fit to the data.
