Conceptualizing a vector space

Question

Sorry for the vague title.

So let's say we have a set $\mathbb R ^2$ that has $+$ and $*$ as laws, how can I show that it is/isn't a $\mathbb R$-vector space if:

$(x,y) + (x', y') = (y+y', x+x') \\ a*(x,y) = (ax,ay)$

What actually is a $\mathbb R$-vector space? Am I actually trying to prove that this is a subspace of $\mathbb R^2$, since we already know that $\mathbb R^2$ is a vector space? Or am I just supposed to go down the list of criteria and check all the requirements?

Truth be told, I'm not totally sure how to interpret this question, and I'm not sure how to conceptualize a vector space. Is it just a space where, if the set fulfills are the requirements, I can stuff vectors into? How would you explain it to a linear algebra newbie? Thank you.

Answer 1

"What actually is a R-vector space? Am I actually trying to prove that this is a subspace of R2, since we already know that R2 is a vector space? Or am I just supposed to go down the list of criteria and check all the requirements?"

I'm assuming it's the second one, because the set you're looking at is exactly $\mathbb R^2$ with its usual operations.

Each individual requirement shouldn't be too hard. For instance, addition is associative because

$ (x, y) + [(x', y') + (x'', y'')] = (x, y) + (x' + x'', y' + y'') $

$ = (x + (x' + x''), y + (y' + y'')) $

$ = ((x + x') + x'', (y + y') + y'') $

$ = [(x, y) + (x', y')] + (x'', y''). $

so that the property is true basically because it's true for real numbers so that the equality in the middle happens. All the rest of them will be basically the same.

As far as how you should conceptualize them, the typical way of thinking of $\mathbb R^2$ is as arrows in the plane. The way of thinking of other vectors space is, honestly, that they all look kind of like $\mathbb R^n$ for some $n$. There are theorems you'll learn that will teach you that all vector spaces over $\mathbb R$ "look like" $\mathbb R^n$.

Answer 2

The notion of a vector space is an algebraic abstraction of the familiar geometric vectors on the plane, or in the 3d space.
Namely (in the first round) we keep only the facts that:

• two vectors can be added
• a vector can be multiplied by a real number

and we write up the essential properties of these operations, which make them analogous to 'addition' and 'multipliciation-by-scalar'.
In contrast, for a $\Bbb Q$-vector space it's only required that vectors can be multiplied by rational numbers.

This generalization turns out to work well enough together with intuitions, e.g. we can specify the dimension of any vector space, and we can introduce coordinates, etc.

As to your specific questions, yes, in an exercise like this you just have to check that the axioms for addition and multiplication-by-scalar are satisfied.
Note that the given operations on $\Bbb R^2$ are just the natural operations, and yes, it is a vector space.