Least Squares Approximation Line for Four Points

Find the least squares approximating line for the set of four points (1, 3), (2, 4), (5, 5), and (6, 10).

The least squares method is a cornerstone of linear regression and statistical analysis, providing an approach to approximate a solution when exact fitting is not possible due to overspecified systems or data noise. In this problem, you are tasked with finding the least squares approximating line for a set of points. This involves finding a line that minimally deviates from the points in terms of the sum of squares of the vertical distances of each point from the line. The solution involves determining parameters, such as the slope and y-intercept, that characterize the best fitting line within the context of linear algebra.

Conceptually, the least squares method applies the principles of linear algebra to solve the problem of minimizing the error of the approximation, which is quantified by the sum of the squares of the residuals. This approach heavily relies on the construction of a normal equation derived from the original data points, and involves solving a system of linear equations, often with the aid of matrices to streamline calculations. One must also understand how to compute the transpose of a matrix and its inverse to solve these kinds of problems efficiently.

Furthermore, the practicality of the least squares method extends to numerous fields, ranging from data fitting in scientific experiments to predictions in machine learning models. Understanding the underlying linear algebra not only improves analytical skills in solving similar problems but also enhances one's capability to apply these skills in real-world scenarios where data fitting is crucial.