91Ó°ÊÓ

A linear equation system consists of multiple linear equations that relate different variables. In this context, we want to fit a plane to certain points using the equation \( y = C + Dt + Ez \). Each point contributes one equation. For example, for a point \((t, z, y)= (1, 1, 3)\), we plug these values into our plane equation to form \( 3 = C + D + E \). This process is repeated for all points, resulting in a system of four equations with three unknowns: \( C, D, \) and \( E \).Since we have more equations than unknowns, we generally can't solve it exactly. However, we can often find the best approximate solution using techniques like least squares, where we aim to minimize the discrepancies between observed and estimated values.

An over-determined system of equations has more equations than unknowns, which means there is generally no exact solution that satisfies all equations simultaneously. In our exercise, we had four equations, coming from four points, trying to solve for three parameters \( C, D, \) and \( E \).### Characteristics of Over-Determined Systems

• More equations than variables.
• Generally no exact solution.
• Solution requires approximating, often using least squares.

Over-determined systems often occur in data fitting problems where you need to find the best compromise solution that 'fits' the data as well as possible. Solving these systems involves minimizing the sum of the squares of the deviations (errors) which occur from trying to fit all of the data points exactly.

Linear regression is a statistical method used for model fitting, where we aim to find the best linear relationship between variables. It's a type of least squares solution, often used when dealing with over-determined systems.In our context, we talk about fitting a plane, which can be thought of as a linear regression problem with more than one independent variable \((t, z)\). The goal is to find coefficients \( C, D, \) and \( E \) that minimize the error between the actual \( y \) values and those predicted by our plane equation.**Benefits of Using Linear Regression**

• Finds the best fit line or plane for given data.
• Minimizes error and provides a clear model for prediction.
• Simple to implement and understand algorithmically.

In our problem, despite the initial system of equations having more equations than unknowns, linear regression allows us to calculate \( C = -1, D = 1, E = 2 \) as the best-fit solution. This provides a way to summarize the relationship among the variables and predict new values based on the fitted model.