Mathematics Hidden Behind Linear Regression

This is about the mathematics that is used in the linear regression (with gradient descent) algorithm. This was a part of my IB HL Mathematics Exploration. Linear Regression is a statistical tool that produces a line of best fit for a given dataset analytically. To produce the regression line manually, one needs to perform operations such as mean-squared error and optimizing the cost function; both are explained in detail later in the document. The main problem arises when the size of the dataset is so large that it becomes computationally inefficient to be done by hand. Therefore, when a dataset size becomes large the computer can perform the task much quicker just with a few simple lines of code in any language. Linear regression algorithm uses a dataset (pairs of input and output values) to generate a line of best fit for that dataset. To start, the algorithm generates a hypothesis in the form??