Linear Transformation of Tetrahedron Rotation Consider the tetrahedron with vertices $(1,1,1), (-1,-1,1), (1,-1,-1)$, and $(-1,1,-1)$. Give the standard matrices for each of the following symmetries of the tetrahedron. Check that each of the matrices is an orthogonal 3∗3 matrix.
a) $+/- 120$ degree rotations about the line joining $(0,0,0)$ and $(1,1,1)$ (vertex)
b) 180 degree rotation about the lines joining $(0,0,1)$ and $(0,0,-1)$
c) Reflection across the plane containing one edge and the midpoint of the other edge (not sure what this means)
I did not learn linear algebra from a geometric perspective and I don't even know where to start. Any advice or links to useful videos are appreciated.
 A: Hint:
Let's say you have a linear transformation $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$.  
For convenience, denote $\mathbf{x}_i = T(\mathbf{e}_i)$, where $\mathbf{e}_i$ is the $i$th standard basis vector.  So, for example, $\mathbf{e}_1 = \langle 1, 0, 0 \rangle$ in $\mathbb{R}^3$.  Likewise, $\mathbf{e}_2 = \langle 0, 1, 0 \rangle$, etc.
Then $T$ is encoded by the $n \times n$ matrix $[\mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_n]$.

As a simple example, consider a $90^\circ$ counterclockwise rotation about the origin in $\mathbb{R}^2$.  Note that $\langle 1, 0 \rangle \mapsto \langle 0, 1 \rangle$, and further $\langle 0, 1 \rangle \mapsto \langle -1, 0 \rangle$.  So our linear transformation is encoded by the matrix $\left[ \begin{array}{ccc}
0 & -1 \\
1 & 0
\end{array} \right]$.

Now applying this to solve your problem will be a bit of work, but if I'm not mistaken, the standard basis vectors lie on the midpoints of edges of this tetrahedron, and edges will map to edges.  It'll be your task to figure out precisely where they are mapping to.

Another approach:
Let $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$, and $\mathbf{v}_4$ denote the vertices of your tetrahedron.  Let $T$ be one of the given linear transformations.  Then $T(\mathbf{v}_1) = \mathbf{v}_i$ for some $i$, and so forth.  In this manner you can arrive at a system of equations whose solution yields the entries of the matrix encoding $T$.
