Background

I currently read a book called Mathematics for Machine Learning and I read chapter 2 which is about Linear Algebra, especially on subchapter 2.8 which is about Affine Space.

The thing is, I learned from the book that affine subspaces are points, lines, and plane in $ \mathbb{R}^{3} $, which don't (necessarily) go through the origin. The affine subspace is defined as $$ L = x_{0} + \lambda b_{1} $$ where:

$L$ is affine subspace

$x_{0}$ is a support vector

$\lambda$ is called parameter (based on book) such that $\lambda \in \mathbb{R}$

$b_{1}$ is directional vector

which is similar to linear regression equation $ y = \beta_{0} + \beta_{1}X_{1}$ (suppose that we only have 1 variable).

Question

Assume that we want to perform linear regression from one variable $X$ to one variable $y$. Then, we got a line equation like:

$$ y = 0.5 + 2X_{1}$$

1. Is the line equation above (which is regression line) is also called an 1-D affine subspace in $ \mathbb{R}^{2}? $.
2. Can we say that whenever we perform a linear regression, we also looking for an (n - 1)-D affine subspace in $ \mathbb{R}^{n} $ that fit well in data?

Thank you :D