Mathematical explanation for Linear Regression working

Suppose we are given a dataset:

Given is a Work vs Experience dataset of a company and the task is to predict the salary of a employee based on his / her work experience. 
This article aims to explain how in reality Linear regression mathematically works when we use a pre-defined function to perform prediction task. 
Let us explore how the stuff works when Linear Regression algorithm gets trained. 

Iteration 1 – In the start, θ₀ and θ₁ values are randomly chosen. Let us suppose, θ₀ = 0 and θ₁ = 0. 

• Predicted values after iteration 1 with Linear regression hypothesis. 

• Cost Function – Error 

• Gradient Descent – Updating θ₀ value 
  Here, j = 0 

• Gradient Descent – Updating θ₁ value 
  Here, j = 1 

Iteration 2 – θ₀ = 0.005 and θ₁ = 0.02657

• Predicted values after iteration 1 with Linear regression hypothesis. 

Now, similar to iteration no. 1 performed above we will again calculate Cost function and update θ_j values using Gradient Descent.
We will keep on iterating until Cost function doesn’t reduce further. At that point, model achieves best θ values. Using these θ values in the model hypothesis will give the best prediction results.