$D_4$ is generated by a rotation $\alpha$ of order 4 and a reflection $\beta$. Its elements $e$, $\alpha$, $\alpha^2$, $\alpha^3$, $\beta$, $\alpha\beta$, $\alpha^2\beta$, $\alpha^3\beta$ give an ordered basis for the algebra $\mathbb{R}[D_4]$.
I have to express the left action of $\alpha$ and $\beta$ as matrices with respect to this basis. I then have to find the invariant one dimensional subspaces for each of these actions.
I think the second part should be straightforward. If $A$ is the matrix representing the left action of $\alpha$, I would just write $Av=v$ and find all such vectors $v$, right? Then $v$ would be invariant. I'm not sure how to do the first part, however, in expressing $\alpha$ and $\beta$ as matrices. I think I should be achieving two $8\times 8$ matrices but I don't know how.
Any help is appreciated!