List some uncommon vector spaces

[benorin]

I'm tutoring a linear algebra/diff eqs class and we are about to start on vector spaces; the point is this, I would like to present them with a variety of unsual vector spaces (along with the usual ones) that they may understand that vectors are not just directed line segments, but rather more than that. Please list some uncommon vector spaces (or common ones with non-standard definitions of scalar multiplication and/or vector addition).

Thanks,
--Ben

 

[Hurkyl]

I suppose you want to stick to real vector spaces, and not vs's over other fields?

Finite-dimensional:
The v.s. of all polynomials of degree less than d.
The v.s. of solutions to a second order ordinary differential equation.
R
C

Infinite-dimensional.
The v.s. of all functions R->R. (or, all continuous functions, or all polynomial functions)



You can obfuscate any vector space you like too; just pick any invertible function f:V -> V, and then the operations

$$s \otimes v := f(s f^{-1}(v))$$
$$v \oplus w := f(f^{-1}(v) + f^{-1}(w))$$

define a new vector space structure on V.

 

[radou]

For example, the set of all converging real sequences. Or, the set of all solutions of a homogenous system of linear equations.

 

[benorin]

Thank you Hurkyl for your excellent examples.
And Thank you radou for the additional example of a v.s., namely the set of all converging real sequences (nice one).

 

[HallsofIvy]

And, although Hurkyl specifically said he was restricting to vector spaces over the real numbers, you can think of the real numbers themselves as a vector space over the rational numbers. That's not only infinite dimensional, it is of uncountable dimension.

Of course, the real numbers form a very simple one dimensional vector space over themselves.

 

[Hurkyl]

(you mean over the rational numbers)

 

[HallsofIvy]

Yes, of course. Thanks, Hurkyl, I'll edit it.

(Of course, the complex numbers can be considered a two dimensional vector space over the real numbers.)

 

[benorin]

The rationals themselves fail to be a subspace of the reals only for the closure under scalar multiplication axiom (was exam question).