Subspace question: prove that a non-trivial subspace of $\mathbb R^2$ must be $\alpha v$ for some fixed v? I need to prove that a non-zero strict subspace of $\mathbb R^2$ must consist of scalar multiples of a fixed vector in $\mathbb R^2$.
Proof: The dimension of the space must be more than 0, but it can't be 2 lest it span $\mathbb R^2$ itself. Hence the dimension must be 1. Spaces of dimension 1 in $\mathbb R^2$ are all of the form $span(\{v\})$ for some $v \in \mathbb R^2$, so the space is the set of all scalar multiples of a single vector.    $  \square$
The proof offered in the Hoffman-Kunze solutions manual is different, so I'm just wondering if my alternate proof is also valid or if it is too "hand-wavey."
 A: When solving textbook exercises, it's good practice to limit yourself to only what theorems and definitions you could access in the pages previous to that question (or in a naturally occurring prerequisite text).
In this case, I believe this is problem 6(b) from section 2.2 of the second edition.  The problem with your proof is that the definitions of "linear [in]dependence," "basis," and "dimension" are all introduced, along with their elementary properties, in section 2.3.  In this regard, your solution is invalid unless you want to pull all the requisite theorems from 2.3 as lemmas and include them in your solution.  I expect that by the time you pull all the definitions and theorems you need from 2.3 in order to make your proof satisfy the above principle, it would end up being much more burdensome than the one given in the solutions manual (though I don't have it to check how likely my expectation is).
That isn't to say that it's never beneficial to use a more advanced tool in exercises.  I recall doing just that in a couple algebra and combinatorics classes, but whenever I did, I preceded the whole assignment with a set of lemmas I needed to prove ahead of time in order to make the methods sufficient to prove what I wanted.
That all said, the harm in using the concept of dimension in this case is purely pedagogical, so it's debatable whether it matters.  However, you will almost certainly eventually find a book that references an exercise within the proof of another theorem (which may occur before or after the exercise).  In that case, you have to be very careful that you don't beg the question, which is the real harm that can occur in this type of situation.

As an afterthought: If your proof is part of an assignment to be graded as part of a class, the actual harm may be that you won't get full points.  Your proof could justifiably be given any grade, from a zero to full points, and unless you have some correspondence with the teacher to use, you won't have an argument against the grade.  You'd just be at the whim of the answer key, and that's not usually a comfortable place to be.
