Learn / Library / Linear algebra

Equations that disagree

Linear algebraHard

The system $x = 1$ , $y = 2$ , $x + y = 4$ has no exact solution. Find its least squares solution, the pair $(x, y)$ that minimises the sum of the squared differences between the left and right sides. What is $y$ ? Give a fraction in lowest terms.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Write the system as a matrix equation and solve the normal equations.

Solution

In matrix form the system has $A = 101011$ and right side $(1, 2, 4)$ , and least squares solves the normal equations with $A^{T} A$ on the left.
Here $A^{T} A = [2112]$ , and applying $A^{T}$ to the right side gives $(1 + 4, 2 + 4) = (5, 6)$ .
So $2 x + y = 5$ and $x + 2 y = 6$ .
Doubling the second equation and subtracting the first gives $3 y = 7$ , so $y = \frac{7}{3}$ .

Answer: $\frac{7}{3}$

Next question Back to the library