Implement prediction, MSE loss, and gradient formulas

From scratch, implement prediction function, MSE loss, and analytic gradients.
Verify gradient correctness using a small numerical gradient check and document the error difference.

Open the repository's Jupyter Notebook and locate the linear-regression implementation area; start by reading the existing model structure and surrounding examples. Implement prediction, MSE loss, and analytic gradients, then verify them with a small numerical gradient check and document the error difference in the notebook.

Written by the indexing model from the issue text.