What is the meaning of $ \mathbb{R}^n$ to $\mathbb{R}^{n+1}$? In linear algebra, what does it mean to go from $\mathbb{R}^1$ to $\mathbb{R}^2$ or $\mathbb{R}^2$ to $\mathbb{R}^3$?

It means to bring a two dimensional space to a three dimensional space, i.e., mapping vectors of the form $(x,y)$ to vectors of the form $(x,y,z).$ This must be done by applying some kind of linear transformation.
