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 $x\in \mathbb{R}$."

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 $f(x+y)$.
