Meaning of $x∈ℝ^d$ notation in matrix approach to regression How does one interpret the $x ∈ ℝ^d$ in the context of regression, see below
in the following context:
I understand the d refers the dimension of the x vector, but I don't get what this notation tells me. Note that I am not familiar with linear algebra, which may explain my confusion.
 A: Mathematically, your understanding is correct, $d$ is the dimension of the vector $x$ and so $x$ exists in the real space of dimension $d$ denoted as $\mathbb{R}^d$. By real space we mean the elements of $x$ are all real numbers. 
In the context of regression, consider the simple case of $d=1$. Let's say we have a linear model for a house price $y$ as a function of its size in square feet $x_1$. Then we just have the linear equation
$$
y = \theta_1 x_1 + b
$$
where $\theta_1$ and $b$ are some model parameters. Now let's say we want to add another variable, the number of bathrooms $x_2$. Then we can write
$$
y = \theta_2 x_2 + \theta_1 x_1 + b
$$
The more variables we add, the longer this equation becomes. So let's write it in matrix form instead. We can define the column vectors
$$
\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ ...\\x_d
\end{bmatrix}\ \in \mathbb{R}^d\ ,\  
\mathbf{\theta} = \begin{bmatrix} \theta_1 \\ \theta_2 \\ ...\\ \theta_d \end{bmatrix}\ \in \mathbb{R}^d
$$
So now we can simply write the linear regression problem as
$$
y = \mathbf{\theta}^T \cdot \mathbf{x} + b
$$
Where we've taken the transpose of $\mathbf{\theta}$ to compute its dot product with $\mathbf{x}$.
TL;DR Matrix form let's us write equations in a simpler form, otherwise things get messy when there are many features and many samples in the input space. The equation may look different however the underlying model is the same. Getting familiar with linear algebra would be valuable.
A: It's needlessly pretentious.  They're saying that $x$ is a $N \times d$ matrix (N rows, d columns), where $N$ is the number of observations and $d$ is the number of variables.  The elements of the matrix are real numbers.
Likewise, they're saying that "y is in the reals" which is to say "y is a vector of real numbers."
I prefer the convention that lower-case italic ($x$) are constants, upper-case ($X$) are vectors, and bold $\mathbf{X}$ are matrices.  It gets harder of course when you get past two dimensions.
I think that a lot of people use pretentious notation in order to demonstrate to others how sophisticated their math is, probably because they're feeling a little bit of imposter syndrome.  
Bottom line:  if you're confused, it could be because the author is really smart and knows things you don't.  More likely however, it's because they can't write clearly or are afraid that you'll understand them and see that their ideas are actually pretty simple.
