Linear Regression - Part(2) [Matrix Calculus]

Previously, I had posted part(1) of the linear regression, where we dealt with machine learning perspective.  And, I had promised to present it with other angles of Mathematics too and here I am!
In this article, we would see it with matrix calculus.

The link to previous article is given below.
http://avidsuraj.blogspot.com/2018/05/linear-algebra-part1-machine-learning.html

I have used the same notation used in the previous article.  So, readers are recommended to read the previous article or at least notation portion of it.

The cost function or error function is written as:
$$ J = \sum  _{i=1}^{m} {{{e}^{(i)}}^{2}}$$
where, ${e}^{(i)}$ is the error for ${i}^{th}$ training example.

Let $E$ represent matrix of all the errors.  So, it is given by:
$$ E = Y-{Y}_{pred} \\
E = Y-(XW+C) \\
E = Y-XW-C $$

Differentiation of a vector, Y with respect to a vector X

Linear Regression - Part(1) [Machine Learning Perspective]

Motivation:

Suppose we want to predict the rent price of a room.  Prediction requires some features or variables on the basis of which we are going to predict.  We think for a while and maybe the rooms which are near to the road cost more.  Similarly, the rooms which have shutters in them are expensive as such rooms are valuable from the commercial perspective.  We also see that big rooms cost more than smaller rooms.  And more and more...

We want to take a handful of those features and based on them, make some prediction regarding rent price.  How could we do it?  How could we model it?  One of the techniques in modeling such rent price model is linear regression and we will discuss it here.