A First Course in Linear Algebra, 2017a

4.11. Orthogonality and the Gram Schmidt Process 249
Example 4.153: Least Squares Solution to a System
Find a least squares solution to the system
⎡ ⎤ ⎡
2 1 [ ]
⎣ −1 3 ⎦ x
= ⎣
y
4 5
2
1
1
⎤
⎦
Solution. First, consider whether there exists a real solution. To do so, set up the augmnented matrix given
by
⎡ ⎤
2 1 2
⎣ −1 3 1 ⎦
4 5 1
The reduced row-echelon form of this augmented matrix is
⎡
1 0
⎤
0
⎣ 0 1 0 ⎦
0 0 1
It follows that there is no real solution to this system. Therefore we wish to find the least squares
solution. The normal equations are
[
2 −1 4
1 3 5
] ⎡ ⎣
2 1
−1 3
4 5
A T A⃗x = A T ⃗y
⎤
[ ] [ ] ⎡
⎦ x 2 −1 4
=
⎣
y 1 3 5
2
1
1
⎤
⎦
andsoweneedtosolvethesystem
[ 21 19
19 35
][ x
y
]
=
[ 7
10
]
This is a familiar exercise and the solution is
[ x
y
]
=
[ 534
7
34
]
♠
Consider another example.
Example 4.154: Least Squares Solution to a System
Find a least squares solution to the system
⎡ ⎤ ⎡
2 1 [ ]
⎣ −1 3 ⎦ x
= ⎣
y
4 5
3
2
9
⎤
⎦