The Least Squares Problem arises when we want to find the best approximate solution to an overdetermined system of linear equations, i.e., a system with more equations than unknowns. This often occurs in real-world applications where data is collected, and we seek a model that best fits the data, even if it doesn't fit perfectly. The goal is to minimize the sum of the squares of the differences between the observed values and the values predicted by the model.
Specifically, given a matrix $A$ and a vector $b$, we want to find a vector $x$ that minimizes $||Ax - b||^2$. The solution $x$ satisfies the normal equations $A^T Ax = A^T b$, where $A^T$ is the transpose of $A$. Solving this system gives us the least squares solution, which provides the closest possible fit to the original system of equations.
Match the following terms with their definitions:
Complete the following paragraph using the words provided: minimize, $A^T A$, $A^T b$, $||Ax - b||^2$, least squares.
The goal of the ___________ problem is to find a vector $x$ that will ___________ the value of ___________. This is achieved by solving the normal equations, which are given by $___________ x = ___________$.
Explain in your own words why the Least Squares Problem is important in real-world applications. Provide an example of a scenario where it might be used and what kind of insights it could provide.
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀