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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?

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IMHO this is a lousy homework question, because the answer is an inherently subjective one: the fact that there is the isomorphism you mentioned means that at some very fundamental level they aren't different, and it feels like bad practice to ask a question like this without at least some guidance as to what you're expecting. –  Steven Stadnicki Jan 24 at 20:37
    
@StevenStadnicki I agree. I interpreted it somewhat like the differentiation between the rational numbers and the reals; the latter embed naturally into the former but they exist in their own right. This is emphasized by the radically different way of defining reals (Cauchy sequences, Dedekind cuts, etc) than of defining rationals. But I can't tell him that because he hasn't had any real analysis. –  Julien Clancy Jan 24 at 20:39
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IMHO it's an excellent homework question. It is important to realize the difference between two things being identical, and two things being isomorphic. To establish what the homework asked, just follow the definition of equality: The two underlying sets are different since you can find an element in one that is not in the other. –  Ittay Weiss Jan 24 at 20:43
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Say we have two identical rooms aside from the number of chairs in them. One room with two seats and another room with 3 seats but the third seat was set on fire and is not usable. While they both serve the purpose of seating any two people, are they exact copies? –  David Peterson Jan 24 at 20:46
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@IttayWeiss That's a fair point; I think there is an excellent homework question in here somewhere. Maybe a better piece of my point would be that they're different in many ways, and the question smacks slightly of having to read the instructor's mind to figure out which one in particular they're thinking about, unless the grading is very generous with respect to differences. –  Steven Stadnicki Jan 24 at 21:32

4 Answers 4

Take any element of one and show that it is not in the other.

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Because there really is only one way to untie a Gordian knot... –  Malice Vidrine Jan 24 at 20:57

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.

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This isn't really a good answer to the question, but +1 because it's really cool. –  Jorge Fernández Jan 24 at 23:36
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@user4140 If "why would it matter" is replaced by the slightly-more-formal "by symmetry", I think this is an excellent answer to the question. It shows concretely why we cannot reason in a certain way, rather than appealing to unnecessary set-theoretic fundamentals and arguing that ordered pairs cannot equal ordered triples. –  you-sir-33433 Jan 30 at 6:09
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@User-33433 I'm afraid seeing "by symmetry" used as a black magic handwaving technique in many years of physics courses has burnt away my ability to use the phrase without flinching, even in appropriate settings. :) thank you for the comment, though. –  Alexander Gruber Jan 31 at 0:46

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.

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I think category theory is a good place to learn about stuff like this, –  Jorge Fernández Jan 24 at 20:55
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I think if you know category theory then you're probably not struggling on the differences between $(a,b)$ and $(a,b,0)$. –  Arkamis Jan 24 at 20:57
    
I agree, but category theory can understand a deeper understanding onto the degree in which things are said to be the same. –  Jorge Fernández Jan 24 at 21:59

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.

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