I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] 978-0-13-187141-0"
In this example, they found a solution (2x1 matrix) of the normal equation:
[a0] = (((C^T)*C)^-1)(C^T)*y [a1]
C is a 5x2 matrix; y is a 5x1 matrix
C = [1 2.60] y = [2.00] [1 2.72] [2.10] [1 2.75] [2.10] [1 2.67] [2.03] [1 2.68] [2.04]
*((C^T)C) is a 2x2 matrix
((C^T)*C) = [5.0000 13.4200] [13.4200 36.0322]
(C^T)*y is a 2x1 matrix
(C^T)*y = [10.2700] [27.5743]
The answer was:
[a0] = [0.056] [a1] = [0.745]
To solve this I think they had to use the formula I listed at the very top, but they did not show work for ((C^T)*C)^-1 (which I guess is the inverse). If someone can please explain with full details of how they solved this normal equation.
I at least understand the given equations I posted, but I don't know why they didn't show the steps of ((C^T)*C)^-1 and how exactly they arrived to:
[a0] = [0.056] [a1] = [0.745] y = 0.056 + 0.745x