How do I convince someone that $\mathbb{R}^2$ and its copy inside $\mathbb{R}^3$ are different? One of my friends is taking a first course in linear algebra now, and one of the problems on his latest homework was to explain why $\mathbb{R}^2$ and $\{(a_1,a_2,a_3) \in \mathbb{R}^3 \mid a_3 = 0\}$ are different. He was confused, and couldn't really come up with an answer. I told him that the latter is somehow similar to the former (the class hasn't covered isomorphisms) but really is composed of different "points" from a different "space". His idea for a proof was to say that you can't add elements of the former to elements of the latter (stemming from the fact that one has a third coordinate and the other doesn't), so they can't be the same space, but this doesn't really get to the nature of things. How should I go about explaining the difference to him?
 A: Let me offer an analogy:
If you look at sets from the structural point of view you could conclude that the only "structure" a set has is the number of elements. In fact if you look at category theory, all sets of the same size are "isomorphic". In other words they are only different in the name of the thing it contains.
So if you only look at the characeristic of the set as a "set" you could say {banana,cherry,apple} is the same as {$x$,$x^2$,$\mathbb R$}. while clearly they aren't the same, the sets have the same structure, the only difference is the objects have different names.
The same happens in your example the vector spaces are the same, the only difference is the name of the elements, proving the elements are distinct, is simple. To show that the spaces are "the same"(also called isomorphic) find a bijection where the operations are preserved.
A: Take any element of one and show that it is not in the other.
A: If $\mathbb{R}^2$ and $$\{(a_1,a_2,a_3)\in\mathbb{R}^3|a_3=0\}$$ are not different, then surely $\mathbb{R}^2$ and $$\{(a_1,a_2,a_3)\in\mathbb{R}^3|a_2=0\}$$ are not different either, because why would it matter which direction we chose to be $z$?  So, you'd expect $$\{(a_1,a_2,a_3)\in\mathbb{R}^3|a_3=0\}\cap \{(a_1,a_2,a_3)\in\mathbb{R}^3|a_2=0\}$$ to be $\mathbb{R}^2$ too, but it should be easy to agree that $\mathbb{R}^2$ isn't a line.
A: Let $A \subseteq \mathbb{R}^3$ denote the subset of interest. Then $A$ basically equals "$\mathbb{R}^2$ + some additional information that tells you how $\mathbb{R}^2$ sits inside $\mathbb{R}^3$."
To emphasize the difference, try asking your friend: "What is the complement of $A$?" Easy question, because we know the context in which $A$ lives; so its complement is $\mathbb{R}^3\setminus A$. Then ask them, what is the complement of $\mathbb{R}^2$? Not such an easy question, that. In fact, the complement of $\mathbb{R}^2$ is undefined.
