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

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.
