What do mathematicians mean by if two vector spaces are isomorphic, then "they are the same"

Question

I have been reading some of the answers here that if two vector fields are isomorphic, then they are "essentially the same".

For example, an answer here:What does "isomorphic" mean in linear algebra?

Because isomorphic vector spaces are the same size and have the same algebraic properties, mathematicians think of them as "the same, for all intents and purposes."

Now when you read it the first time, it is pretty impressive. But when you are sitting down and doing a problem. You are wondering just exactly what it means. What algebraic properties are the same? What do you mean by size? What does it mean by "the same, for all intents and purposes". Can someone please concretify this notion using the following example?

Claim: An isomorphism between symmetric matrices $S^{n \times n}$ and $R^{n^2}$ is given by map $F = vectorize(A)$

Where given $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, vectorize(A) gives $\begin{bmatrix} a_{11} \\ a_{12} \\ a_{21}\\ a_{22} \end{bmatrix}$

Ok, this is a nice math trick. But so what? Ok so there is an invertible map. What does that imply about the algebraic properties of the two vector spaces?

Answer 1

Examples of isomorphic vector spaces:

• $V_{xy} = \{(x, y, 0) : x, y \in \Bbb R\} \subseteq \Bbb R^3$

• $V_{xz} = \{(x, 0, z) : x, z \in \Bbb R\} \subseteq \Bbb R^3$

• $V_{yz} = \{(0, y, z) : y, z \in \Bbb R\} \subseteq \Bbb R^3$

You wrote:

Ok, this is a nice math trick. But so what? Ok so there is an invertible map.

That's the spirit! Are you amazed and dumb-founded that all of these planes are "the same" vector space? No, of course not! And you shouldn't be! Why should it matter what two-dimensional subspace of $\Bbb R^3$ we're looking at? It shouldn't; they're all just $\Bbb R^2$ in disguise! And the same with one-dimensional subspaces (except they're $\Bbb R$ in disguise). This is isomorphism at its finest.

Yes, "isomorphism" is a fancy word. But the idea isn't, it is just "sameness" codified. Isomorphisms are usually pretty boring, because of the essential sameness. There's no need to think isomorphism is the best thing since sliced bread.

Or get excited, that's good too! Should someone who collects baseball cards work to acquire multiple copies of every card? Of course not, because they're all "the same" (isomorphic as baseball cards). So too with mathematicians and vector spaces; if you like less work (free understanding once an isomorphism is exhibited) you should like isomorphism.

Sometimes it isn't immediately as boring. Ordered pairs of real numbers, complex numbers, and expressions of the form $a + bx$ are all "the same" (isomorphic as two-dimensional vector spaces over $\Bbb R$), if we only add and multiply by real numbers. But we usually learn about linear expressions first, then complex numbers, then the vector-space structure (not just as points, but things that can be added and scaled) of the $xy$-plane. Some people handle them all differently until they realize they're all "the same" (some people never reach this point) and some people intuitively grasp it before (if ever) hearing about isomorphisms.

Fancier examples exist (binary numbers and subsets, for example).

Answer 2

A vector space is a set of vectors, $V$, associated with a set of scalars, $S$, with operations of vector addition (a function $V\times V \to V$) and scalar multiplication (a function $S\times V\to V$), which satisfy certain axioms. That is all of the structure that is relevant when you are just talking about a vector space.

Two vector spaces $V$ and $V'$ (both associated with scalars $S$) are isomorphic as vector spaces (isomorphic when just considering their bare vector space structure) when:

1. There is a bijective mapping $f:V\to V'$ (it is one-to-one and its inverse is one-to-one).
2. The mapping $f$ preserves the vector space structure, so a) $f(a+b)=f(a)+f(b)$ and b)$f(ka)=kf(a)$ where $a$ and $b$ are vectors in $V$ and $k$ is a scalar.

So an isomorphism takes one vector space and transforms it into another, in such a way that every element of one has a unique corresponding element in the other and vice versa, and all of the relevant structure that makes it a vector space is preserved. So you can do your operations or calculations in the first vector space; or you can transform to the second vector space, do them there, and transform back. It doesn't matter because the results are the same.

This is the sense in which isomorphic vector spaces are the "same space": each element in one has its own image in the other, and the two spaces behave (as vector spaces) in exactly the same way.

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In general you can talk about different kinds of isomorphisms which preserve different kinds of structures on spaces. If you add more structure to the vector spaces, such as an inner product to turn them into inner product spaces, then two spaces which are isomorphic as vector spaces may or may not be isomorphic as inner product spaces. It all depends on the structures that are defined. Take a look at category theory if this is interesting to you. Hope that helps!