In the book Hands-On Machine Learning with Scikit-Learn & TensorFlow, the author only showed the formula for the Batch Gradient Descent method, such as:
$ \dfrac{\partial}{\partial \theta_{j}} MSE(\theta)= \dfrac{2}{m}\sum_{i=1}^{m}(\theta^T \cdot \boldsymbol{x}^{(i)}-y^{(i)})\cdot x^{(i)}$
So that the gradient vector of the cost function is: $\bigtriangledown_{\theta}MSE(\theta) = \begin{bmatrix} \dfrac{\partial}{\partial \theta_{0}} MSE(\theta_0) \\ \dfrac{\partial}{\partial \theta_{1}} MSE(\theta_1) \\ \dfrac{\partial}{\partial \theta_{2}} MSE(\theta_2) \\ \vdots \\ \dfrac{\partial}{\partial \theta_{n}} MSE(\theta_n) \end{bmatrix} = \dfrac{2}{m} \cdot X^T \cdot (X \cdot \theta - y)$
The MSE cost function is defined as: $MSE(\theta) = \dfrac{1}{m}\sum_{i=1}^{m}(\theta^T \cdot \boldsymbol{x}^{(i)}-y^{(i)})^2$
Is there anyway who could kindly step by step show me the proof of the cost function's gradient vector formula (using linear algebra) above?