Definitions of vector space and subspace

[elementbrdr]

I am using Axler's Linear Algebra Done Right as a text for independent study of linear algebra. Axler basically defined a vector space to be a set which has defined operations of addition and multiplication (and which comports with certain algebraic properties) and that contains an additive identity (which I understand to mean essentially {0}). As I understand it, a subspace is simply a subset of a vector space that has the same properties.

I am having trouble understanding what this all means for sets other than $$F^n$$ I find vector space and subspace to be intuitive concepts when applied to complex number sets (particularly so with respect to real number sets). But I have trouble understanding what vector space and subspace actually means when applied to sets containing non-numeric elements. For example, Axler discusses subspaces in the context of the set P(F), which is the set of all polynomials with coefficients in F, and the function p(x), which is a polynomial function. The example he provides is that the set $$\{\rho \in P(F) : \rho(3) = 0\}$$ is a subspace of $$P(F)$$
This makes a bit of sense to me, but I'm having trouble understanding its significance. I don't get why the concept of subspace is useful for anything other than vector space having ordered n-tuples as elements. I'm having similar difficulty with the concept of direct sum, but I'll save that until after I've cleared up my current confusion.

Thanks.

 

[micromass]

Hi elementbrdr!

It seems that you understand quite well what a subspace is, so you're only asking yourself what the use is of subspaces?

Well, subspaces are just a formulation of some handy properties. That is, it's just saying that it's a vector space in it's own right (with operations that coincide with that of the larger vector space). You will see the term subspace used time after time in linear algebra, so you will soon see it's importance.

Also, if you understand the vector space Fⁿ, then you actually understand them all because all vector spaces have this form!

 

[espen180]

P(F) is a little special, because it is an infinite-dimensional vector space.

Have you studied bases of vector spaces yet? Do you agree that every vector in an n-dimensional vactor space, given a basis, is in one-to-one correspendance with its coordinate vector? This relationship let's us represent any n-dimensional vector space as an ordered n-tuple in a basis.

I think the significance of subspaces and direct sums will become evident when you study linear transformations.

 

[elementbrdr]

Thanks for the responses!

espen180, I have not yet studied bases of vector spaces. As of now, I have only advanced as far as learning span.

I think the takeaway here is that I should be patient and this will all make sense in due time :)

 

[Studiot]

Axler basically defined a vector space to be a set which has defined operations of addition and multiplication (and which comports with certain algebraic properties)

I don't know Axler, but I doubt he says that a vector space is only a single set.
You actually need two sets to define scalar multiplication, which I presume you mean since scalar multiplication is fundamental to all vector spaces, whilst vector multiplication is not.

You have your field set - the scalars and your vector set - the vectors.
Together they make the vector space.