QUICKSTART.md

Quick Start

This guide explains how to quickly run and understand the Normal Equations solver (scalar form) for simple linear regression.

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🚀 1. Clone the Repository

git clone https://github.com/USERNAME/Normal-equations-scalar-form-solver-course.git
cd Normal-equations-scalar-form-solver-course

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🧠 2. Mathematical Background

We solve the simple linear regression problem:

$$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i $$

in least squares form:

$$ \min_{\beta_0, \beta_1} \sum_{i=1}^{n} (y_i - \beta_0 - \beta_1 x_i)^2 $$

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🔎 3. Closed-Form Solution

Using the normal equations:

$$ (X^T X)\beta = X^T y $$

The solution is:

$$ \beta = (X^T X)^{-1} X^T y $$

For scalar regression, the slope and intercept can be written explicitly:

$$ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})} {\sum (x_i - \bar{x})^2} $$

$$ \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} $$

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💻 4. Run the Solver

Install dependencies:

pip install -r requirements.txt

Run the example:

python example.py

Or import the solver in Python:

from solver import normal_equation_scalar

beta0, beta1 = normal_equation_scalar(x, y)

print(beta0, beta1)

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📊 5. Visualization (If Included)

If the project contains visualization:

python visualize.py

You should see:

• Data points
• Regression line
• Fitted model

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🧪 6. Run Tests

If tests are included:

pytest

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🎯 What You Learned

After running this project, you understand:

✔ How normal equations are derived ✔ How closed-form least squares works ✔ Why matrix inversion appears ✔ How scalar form relates to matrix form

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