Prove reflection in a hyperplane is a linear map 
Let $\alpha \in \mathbb{R}^n$, $n \geq 2$, be a non-zero vector. Define a reflection in the hyperplane perpendicular to $\alpha$ by:
  $$\sigma_{\alpha}(v) = v - \dfrac{2(v, \alpha)}{(\alpha, \alpha)} \cdot \alpha$$
  ($(x, y)$ is the usual inner product on $\mathbb{R}^n$).
1) Show $\sigma_{\alpha}$ is a linear map that fixes the hyperplane orthogonal to $\alpha$ and sends $\alpha$ to $-\alpha$.
2) Given $\alpha, \beta$ non-zero vectors, determine when the subgroup $\langle \sigma_{\alpha}, \sigma_{\beta} \rangle$ is infinite. Find its order when it is finite.

For 2) I don't understand what the group is. If $\sigma_{\alpha}$ and $\sigma_{\beta}$ are elements of a group, what other elements do they generate? Like for example, $\sigma_{\alpha}(\sigma_{\beta}(v)) = \left(v - \dfrac{2(v, \beta)}{(\beta, \beta)} \cdot \beta \right) - \dfrac{2\left(v - \dfrac{2(v, \beta)}{(\beta, \beta)} \cdot \beta, \beta \right)}{(\beta, \beta)} \cdot \beta$ which I guess makes sense (in the sense that dot products work in this function since the dot product is between vectors). But how do I know when there will be an infinite many number of these, and when there will be finitely many?
I can't even find an identity function $\sigma$, because a composition of $\sigma_{\alpha}$ and $\sigma_{\beta}$ is $\sigma_{\beta}$ only when $\sigma_{\alpha} = v$, but this is a constant function and does not reflect $\alpha$ about the hyperplane to $-\alpha$, so this constant function cannot be in the group.
 A: For your first subquestion of 1), the hyperplane is described in the question: it is the hyperplane orthogonal to $\alpha$. You know from linear algebra that this hyperplane is the solution of the equation $\alpha \cdot v = 0$. So your goal is to take any $v$ in that hyperplane, i.e. take any $\nu$ such that $\alpha\cdot v=0$, and prove the equation $\sigma_\alpha(v)=v$. 
For your second subquestion of 1), $\alpha$ is a vector, and it is a constant, so it is a constant vector (unlike $v$ which is a vector, and it is a variable, so it is a variable vector). To say that a function $f$ sends $a$ to $b$ means $f(a)=b$. So to say that the function $\sigma_\alpha$ sends $\alpha$ to $-\alpha$ means that $\sigma_\alpha(\alpha)=-\alpha$. That's the equation you are asked to prove.
For 2), you are correct that a function is not a group, but the question does not ask you to believe that a function is a group. Instead, the question asks you to believe that the set of all linear isomorphisms of $\mathbb{R}^n$ is a group under the binary operation of composition --- you may have heard of this group, it is denoted $GL(n,\mathbb{R})$. Also, you are asked to believe that if $\alpha$ is a constant vector then $\sigma_\alpha$ is an element of the group $GL(n,\mathbb{R})$. Also, if you fix two constant vectors $\alpha$ and $\beta$, then you are asked to believe that there is a subgroup of $GL(n,\mathbb{R})$ denoted $\langle \sigma_\alpha,\sigma_\beta \rangle$ and called the subgroup of $GL(n,\mathbb{R})$ that is generated by $\sigma_\alpha,\sigma_\beta$.
A: Get some intuition from three dimensions first. Say the intersection of the two planes is the axis spanned by $\gamma$. Then $\{\alpha,\beta,\gamma\}$ is a basis, and the reflections only act on the $\alpha$ and $\beta$ components of any vector. This generalizes: prove that ${\rm span}\{\alpha,\beta\}$ is the orthogonal complement of the planes' intersection.
(More generally, $A^\perp\cap B^\perp=(A+B)^\perp$ for any subspaces $A,B$ of an inner product space.)
So really, you only need to worry about what the reflections do to the plane $\alpha$ and $\beta$ generate. That's only two dimensions to worry about. Without loss of generality, say one reflection is across the $x$-axis and the other is across the line $y=\tan(\theta)x$ (which makes an angle of $\theta$ with the $x$-axis). What exactly is the composition of the two reflections then?
If it helps, draw these two lines on a piece of paper, and put a point $P$ just under the $x$-axis in the fourth quadrant. Reflect across the $x$-axis to get a point $Q$, then reflect across the other line to get point $R$. If you label all of the angles made (between the lines and the imaginary line segments joining the origin to the three points) you should be able to make some deductions about the angles, and then get an idea for what the composition of the two reflections is.
(Spoiler: you'll be thinking about $n$-gons and dihedral groups soon after that.)
