Is a vector in $\Bbb R^2$ equivalent to a vector in $\Bbb R^3$ with the same first two components and $0$ as the third component?

Question

For example would $\langle u_1, u_2 \rangle$ be equivalent to $\langle u_1, u_2, 0 \rangle$ even though the former is in $\Bbb R^2$ and the latter in $\Bbb R^3$? I know these are visually the same; however, does defining the vector in $\Bbb R^3$ lead to any additional properties?

Answer 1

Almost. The two vector spaces are isomorphic, meaning that, there is an invertible relation between the two spaces. Given a vector from any one of these 2 spaces, there exists an unique corresponding vector in the other space. Thus these spaces not the same, but have the same properties.

The same is true for points on the real line in $\mathbb{R}$ and points $(x,y)$ in $\mathbb{R}^2$ with $y=0$.

Answer 2

Yes. We have the isomorphism $$\begin{matrix}\mathbb{R}^{2}&\longrightarrow&\mathbb{R}^{2}\times\{0\}\\(x,y)&\longmapsto&(x,y,0)\end{matrix}$$ and so we can see $\mathbb{R}^{2}$ as a subspace of $\mathbb{R}^{3}$.