Finding Closest Vector Using Least Squares Approximation

Find the vector $\hat{x}$ such that $A\hat{x}$ is the closest to $b$ using the least squares approximation.

In this problem, you're dealing with an important concept in linear algebra known as the least squares approximation. This approach is widely used when you want to solve an overdetermined system of equations—that is, more equations than unknowns—when a perfect solution may not exist. Least squares method helps you to find the solution that minimizes the sum of the squares of the residuals, which are the differences between observed and computed values.

The task is to find a vector that best aligns with your target vector after being transformed by matrix multiplication. This is particularly useful in data fitting, where you have a model that should best fit a set of observations. From a geometrical perspective, you're projecting a vector onto a subspace spanned by the columns of the matrix, which gives the best approximations possible.

Understanding how to apply the least squares approximation can be incredibly useful in various fields such as statistics, data science, and machine learning, where the ability to analyze and interpret data is crucial. Familiarity with concepts like the normal equations, orthogonality, and projections is key to mastering problems involving least squares.