teaching concept of “subspace” to linear algebra students

I'm currently a T.A. for an introductory combined linear algebra & differential equations course geared toward engineering students. One major problem I've run across is that most of my students, when asked determine for instance whether the set $S=\left\{f\colon \mathbb{R}\to\mathbb{R}: f(x)=f(-x)\text{ for all }x\in \mathbb{R}\right\}$ is a subspace of the space $V$ of all real-valued functions, will begin with some variation of:

Let $x,y\in S$. Then $f(x+y)=\ldots$

On the other hand, when asked to do the same thing for something like $S=\{\mathbf{x}\in \mathbb{R}^n:A\mathbf{x}=\mathbf{b}\}$ (where $A$ is an $m\times n$ matrix and $\mathbf{b}\in \mathbb{R}^m$) and $V=\mathbb{R}^n$, these same students are able to proceed correctly.

What are some good ways to address this sort of issue?

EDIT (4:24pm 3/15/2013): Note that for the first $S$ above, the students were actually given a description of the set in prose-form, i.e. "Let $S$ be the set of real-valued functions $f$ on $\mathbb{R}$ such that $f(x)=f(-x)$ for all $x\in \mathbb{R}$."

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combined linear algebra & differential equations Do you mind if I ask how that works? –  Jonathan Rich Mar 15 '13 at 19:57
@JonathanRich Solving Linear ODE's uses a lot of stuff from linear algebra. –  Avi Steiner Mar 15 '13 at 19:58
That's how it was at my university, as well. It was LA lite though. The little I remember included "just take my word for it that these solutions are linearly independent," det -> eigenvalue/vectors -> solutions of linear systems of diffeq –  Tyler Mar 15 '13 at 20:04

Sounds like they are struggling with the jump from a concrete vector space like $\mathbb R^n$ to an abstract vector space. If I were you, I would run a lesson/review session or write a worksheet that

1. States the defining properties of a vector space
2. Encourages the students to think of sets that satisfy these properties which are not tuples of numbers

I would discuss polynomials, functions, and subspaces thereof in depth with them, being sure to lead them to examples and counterexamples.

If the only thing in my power was to revise the question, I would create a part (a) asking them to prove the space of real functions was a vector space (in prose, rather than giving them symbols to start playing with before thinking about it) and maybe a part (b) asking them to name a few elements of the set you will ask them about in (c) your original question.

IMHO, since sounds like a beginning-intermediate course that isn't aiming to train math majors, don't make them work so hard to parse out the mathematical ideas. Specifically, I think they could be getting distracted by the set builder notation. I would just call this "the set of even functions" here.

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I also think that students have a lot of trouble with set builder notation. The courses that use it very seldom define it or let the students work with it long enough for them to be well accustomed to it. –  Raskolnikov Mar 15 '13 at 20:17
@orlandpm See my edit. –  Avi Steiner Mar 15 '13 at 21:28
@AviSteiner noted. In addition to leading with suggestive part (a) and (b) you might want to write "(HINT: the sum of two functions is the usual $f+g$ and scalar multiplication of $f$ by real $r$ is just $rf$)" This at least hints that they should be doing algebra on the functions, not their arguments. –  orlandpm Mar 15 '13 at 22:08

It sounds like they have a fixed prototype of a vector space as looking like $\Bbb R^n$. Sure, that is true, but I think they are so attached to the image of $\Bbb R^n$ that they aren't recognizing vector spaces that appear in other forms.

When you ask them to think abstractly about "the vector space of functions from $\Bbb R ^n\rightarrow \Bbb R$" their prototype is breaking down. In all likelihood, their image of functions is that functions start on $\Bbb R^n$ and go somewhere, and the idea that functions as entities could be added and scaled as vectors is new.

Spend a little more time trying to convince them that the functions are vectors in a vector space, and hopefully that will remind them that what they should be checking is $f+g$ instead of $f(x+y)$.

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I run into the same problem with my students and I think the best thing you can do is precsiely what you already do: assign them a problem like that and then point out why they are doing it wrongly. But let them think about it first.

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