TITLE: What is the difference between a 2-dimensional plane in $R^3$ and $R^2$?
QUESTION [8 upvotes]: A $3\times 3$ matrix with $2$ independent vectors will span a $2$ dimensional plane in $\Bbb R^3$ but that plane is not $\Bbb R^2$. Is it just nomenclature or does $\Bbb R^2$ have some additional properties that other planes don't?

REPLY [3 votes]: This is an interesting question. The usual definition of $\mathbb{R}^2$ is the set of all ordered pairs of real numbers, with pointwise operations. With this definition, no subspace of $\mathbb{R}^3$ is literally $\mathbb{R}^2$, although the $xy$ plane is quite similar, as we only need to adjoin an extra $0$ to the end of each sequence in $\mathbb{R}^2$. 
There is a different viewpoint, however, which is more structural, and has a good analogy with category-theoretic foundations. Some mathematicians are uncomfortable with "extra" set theoretic baggage in the definitions of mathematical objects. For example, some dislike the fact that, in the usual definition of the natural numbers, we have $1 \in 5$. This is because the $\in$ (set membership) relation does not seem like an appropriate relation for natural numbers.
Similarly, the "take the first coordinate of the vector $v$" operation is not on its own a natural operation on a vector. The axioms of a vector space do not include anything about coordinates. Instead, thinking structurally, we should say "given an ordered basis $b_1, \ldots, b_n$, take the component of the vector $v$ in the direction of $b_1$". From this viewpoint, all isomorphic copies of $\mathbb{R}^2$ are equally good. Only by "looking inside" could we distinguish the "usual" $\mathbb{R}^2$ from its isomorphic copies. So from this structural viewpoint, any plane in $\mathbb{R}^3$ through the origin is a perfectly good copy of $\mathbb{R}^2$.
I should emphasize that the structural viewpoint is not commonly presented in elementary linear algebra texts. This leads to some challenges in seeing the structural viewpoint.  For example, the dot product of two vectors in $\mathbb{R}^2$ should also only be defined relative to a basis - you know you have understood the structural viewpoint when you can grok the sentence "the dot product is an operation on pairs of finite sequences of real numbers, not on pairs of vectors".