Brief review in linear algebra - Festina lente (in English)

We may encounter linear algebra in considering data analysis, because there are lots of occasions to treat physical amounts like length, temperatures etc, and we will treat them with regression analysis or other statistical instruments. Those numbers are measured in real number $\mathbb{R}$ , and $\mathbb{R}^n$ , which is a combination of the real numbers.

From my point of view, it would be very important to think about brief "mock" models before investigating them further. This kind of "mock" models contain the essential idea, or if we don't construct even "mock" model, it might mean that we don't understand the whole picture of it.

Let's think about the "mock" model for linear algebra. What do you think of it?

My answer is simply "Line", which describes like
$y=ax+b$
where $a,b$ are the constants in $\mathbb{R}$ and $x,y$ variables in $\mathbb{R}$ . We learn it as geometrical equation in 2-dimensional plane $\mathbb{R}^2$ in general, but I'd like to consider it as the linear transformation defined as follows:
$\mathbb{R}^1\ni x \mapsto y(=ax+b)\in\mathbb{R}^1$

In some situation, it is important to throw away the idea in order to understand the essence. In the example above, we have to forget the 2-dimensional plane, at first, i.e. we'd like to focus on the linear transformation in considering the equation $y=ax+b$