Determining transition matrix I've been stuck on this problem without any idea of how to proceed. I'm doing some Linear Algebra problems on my own so I don't have explanations to these problems (but there are solutions). 
Consider the map $A$ of $\mathbb R^3$  into itself given the mapping
$Ar = r - 2(n \cdot r)n$ where $r = (x,y,z)$ and $n=(2/3, 1/3, 2/3)$.
Compute $A^2$. 
Where should I start? Taking an undergrad linear/diff-eq class.
 A: Edit: Seems I have jumped the gun a bit, the method in my first try to answer the question is rather tedious. Lets look what $A$ does to $\bf{n}$:
$$
A\left(\bf{n}\right)=\bf{n} - 2\left( \bf{n} \cdot \bf{n} \right) \bf{n}\\
= -\bf{n} 
$$
while it keeps vectors perpendicular to $\bf{n}$ invariant, because $\left(\bf{n} \cdot \bf{r} \right)=0$ if $\bf{r}$ is perpendicular to $\bf{n}$. So $A$ mirrors $\mathbb{R}^3$ on the plane perpendicular to $\bf{n}$ and therefore if you apply it twice you end up where you started. You can show this, by substituting $A\left(A\bf{r}\right)= A\bf{r} - 2 \left( \bf{n} \cdot \left(A\bf{r}\right)\right)$ and then substituting the definition again and using the linearity of the scalar product. 

My old answer:
You should write down the equation in components. (Try this first, before reading the rest of the answer.) 
You can compute the matrix of any linear transformation by noticing that a linear transformation is linear. For a vector $\bf{x} \in \mathbb{R}^n$ and a basis $\bf{e}_{k \le n}$ of $\mathbb{R}^n$ you have $\bf{x} = \sum_{k=1}^{n} \alpha_k \bf{e}_k$ and for a linear transformation $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ 
$$ 
L\left(\bf{x}\right)= \sum_{k=1}^{n} a_k L\left(\bf{e}_k\right)
$$
if you then plug in the canonical basis, you can directly read of the matrix. Then you can calculate $A^2$. This directly generalizes to non automorphisms and you should write it down explicitly, it really clears up a lot about matrices.
