Is it too much rigor to turn a set into a vector space?

Question

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point.

The example went along the line of something like:

Given a set $\mathbb{X} = \mathbb{R^2}$, let $x = (u_1,u_2), u_1,u_2\in \mathbb{R}$

Define $(u_1,u_2)$ + $(v_1,v_2)$ = $(u_1+v_1, u_2+v_2)$, $\alpha (u_1,u_2) = (\alpha u_1, \alpha u_2)$

Then $\mathbb{X} = \mathbb{R^2}$ becomes a vector space over $\mathbb{R}$

... Is it really necessary to construct a vector space out of the set of $\mathbb{R^2}$? Can't we simply just use $\mathbb{R^2}$ as a vector space without forcing ourselves to define operations on $\mathbb{R^2}$ everytime we wish to treat it as a vector space?

Is this too much rigor or is a standard practice?

Answer 1

When people say $\mathbb{R}^2$, it is commonly implicit all its "canonical" structure (as a vector space, topological space, metric space etc).

But you must know what this canonical structure is!

Imagine you enter a room where everyone knows Bob. You also know Bob. But everyone also knows Bob's dog, which everyone simply assumes everybody else knows: it is a simple dog, after all! It so happens that you've never been introduced to Bob's dog, nor seen in photos etc. You only know that it is a dog. You can even say: Oh, okay, Bob has a dog. But you don't know its color, or if it eats a lot: you don't know its characteristics/structure. What will you do if you need to buy a present for Bob's dog? No matter how simple it is, you still need to know Bob's dog to know its "structure". After you've been introduced, you can enter the "club", and talk about Bob, with an underlying, well-known dog.

Answer 2

There is never too much rigor in math. ;)

A vector space is a set of vectors over a set of scalars, with two operations: vector addition and scalar multiplication. If you don't have operations, you don't have a vector space.

And you will discuss different operations on $\Bbb{R}^2$ other than the standard addition and multiplication, so yes it is necessary to be explicit with what operations you are talking about.

Answer 3

The notation $\Bbb R^2$ already refers to the vector space you're referring to, so there's no need to define the vector space structure every time you use the notation. But we do need to define everything, even common notation, at least once in our lives so that terms even have meaning. Whether or not you need to rewrite the definition again in another context depends on your purposes, the audience, and the how standard your terms or notations are. In other words, just like words themselves.

Answer 4

To use language effectively, you have to consider your audience...

In the context of an introductory upper-division course on linear algebra, it's good to be explicit about the vector space operations. After all, they do have to be defined, and there's value in emphasizing that point.

In a professional context, it would be insane. The reader knows what you mean if you call $\mathbb R^2$ a vector space. Acting as if they didn't would be an insult and a waste of time.

Answer 5

Edit: As someone pointed out this is not a vector space as it violates one of the properties but it was made on the top of my head so I didn't check.

$\mathbb{R}^2$ simply means the cartesian product of $\mathbb{R}\times\mathbb{R}$, which is the set of all ordered pairs $(a,b)$ with $a,b\in\mathbb{R}$. This is not a vector space, this is not a metric space, itis nothing but a set of ordered pairs. You must always define what operations means for it to be meaningful. What does $a+b$ mean for $a,b\in\mathbb{R}^2$? Unelss defined we don't know, you can't make assumptions that are unwarrented but must clearly define it which we do most often by utilizing previous structures where it is already defined. That is why you must define it, I can for example define a vector space like $$(a,b)+(c,d)=(2a+2c,b+d)$$ and $$a(b,d)=(2ab,ad)$$ and so on, it'll be peculiar but it'll work and we don't know it unless stated.

Answer 6

So most of us encounter $\mathbb R^2$ as the cartesian plane in which we create axes, sketch functions, explore geometry, encounter vectors etc. It is a familiar object.

Then we meet, in algebra, the abstract concept of a vector space. We understand the idea - one thing we want to do is to take what we've learned in $\mathbb R^2$ and other examples which are familiar to us, and to use that knowledge in a more general context. We perhaps want to explore more dimensions, and need to know whether there are things which are special about two dimensions which won't apply in three. We encounter the idea of complex vector spaces, and someone perhaps tells us that the whole idea has proved fruitful in abstract algebra.

So we make a definition in abstract. But does the definition in fact relate to our original and most familiar example? Will it help us to make the conceptual leap from the concrete to the abstract? Let's check that it works ... and then we're ready to go.

As others have said, we don't need to do it every time - but we do need to do it at least once.

Answer 7

This is Mark Bennet's answer, with a different emphasis. I am also operating under the assumption, like many others, that you saw this toward the beginning of a set of linear algebra, abstract algebra, or analysis notes. I agree with you that the check that $\Bbb R^2$ is a vector space is generally boring, obvious, and tedious, but I posit that, in this context, it still has value.

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One reason that we go through the rigor of showing that $\Bbb R^2$ is a vector space is to provide some evidence that a vector space "is what you think it is". Or, if you don't think anything of it, to give you some idea of what you should think it is

A vector space is, after all, an object with two operations satisfying nine axioms, and an additional object called a field which is an object with two operations satisfying ten axioms. (Numbers varying depending on how you count, of course). But it's very easy to read this list and then come out at the end without having any clue what the heck a vector space is.

A very good reason that we go through the (boring, tedious, and obvious) effort of showing that $\Bbb R^2$ is a vector space is because we need some indication that these axioms have any sort of relationship to any object we care anything about. Most likely, the author is assuming you know what $\Bbb R^2$ is, and how to add vectors, and so on, and is using that past knowledge to suggest that vector spaces in general might be something that you should care about as well.