Question

Find the least-squares solution

x

∗
of the system
⎣
⎡
​

1
−1
3
​

−1
1
5
​

⎦
⎤
​

x
=
⎣
⎡
​

0
10
9
​

⎦
⎤
​

x

∗
=[

Answer 1

To find the least-squares solution x*, we need to find the matrix A^T A and the vector A^T b, where A is the coefficient matrix and b is the constant vector in the system of equations Ax = b. Then we can solve the normal equations (A^T A)x* = A^T b to find the least-squares solution.

A = ⎣⎡1 -1 3⎦⎤, b = ⎣⎡0 10 9⎦⎤

A^T A = ⎣⎡1 -1 3⎦⎤ ⎡⎣1 -1 3⎤⎦ = ⎣⎡3 -3 12⎦⎤

⎡⎣-1 1 5⎦⎤ ⎣⎡-3 3 -12⎦⎤

A^T b = ⎡⎣1 -1 3⎦⎤ ⎡⎣0⎦⎤ = ⎡⎣-10⎦⎤

⎡⎣-1 1 5⎦⎤ ⎣⎡10⎦⎤ ⎣⎡-9⎦⎤

Now we can solve the normal equations (A^T A)x* = A^T b:

⎡⎣3 -3 12⎦⎤ ⎡⎣x1⎦⎤ ⎡⎣-10⎦⎤

⎢⎣-3 3 -12⎦⎥ ⎢⎣x2⎦⎥ = ⎢⎣-9⎦⎤

⎣⎡12 -12 48⎤⎦ ⎣⎡x3⎦⎤ ⎣⎡30⎦⎦

We can solve this system of equations using row reduction:

R2 = R2 + R1

R3 = R3 - 4R1