1. The problem: Understand when to apply least squares approximation in matrices. 2. Least squares approximation is used when you have an overdetermined system of linear equations, meaning there are more equations than unknowns, and no exact solution exists. 3. The goal is to find an approximate solution that minimizes the sum of the squares of the residuals (differences between observed and predicted values). 4. The formula used is: $$\hat{x} = (A^T A)^{-1} A^T b$$ where $A$ is the matrix of coefficients, $b$ is the vector of observations, and $\hat{x}$ is the vector of estimated parameters. 5. Important rules: - $A^T A$ must be invertible. - The method projects $b$ onto the column space of $A$. 6. We apply least squares approximation when: - The system $Ax = b$ has no exact solution. - We want the best approximate solution in the least squares sense. - Data fitting problems, regression analysis, and signal processing. 7. In summary, least squares approximation is applied to find the best fit solution to inconsistent or overdetermined linear systems using matrix operations.