How would I find this eigenvalue? I'm told to let $A$ be the matrix of the linear transformation $T$ and without writing $A$, find an eigenvalue of $A$ and describe the eigenspace. The first is to let $T$ be the transformation on $\mathbb R^2$ that reflects points across some line through the origin. Linear transformations always confuse me so I'm not really sure how to go about doing this. 

I thought that possibly the standard matrix of a reflection across the origin would be our $A$ and then we find the eigenvalue from that. Is that somewhere in the right direction?
Since you are asked to work in $\mathbb{R}^{2}$ so geometric thinking will be of immense help. Note that you are asked to find the eigen-value of the matrix of reflection across some line "passing" through the origin not "about" the origin. You can easily see the difference by drawing some points in $xy$ plane and the reflecting them about the origin and then across some line passing through the origin. 
\Now recall that a linear transformation from some vector space to another is completely determined by its action on the basis of the domain vector space. So, in particular you can see the reflections of the standard basis of $\mathbb{R}^{2}$ across the given line and then you will be able to determine the transformation. Write the matrix of that transformation and then eigenvalue(s) can be found easily. For understanding point of view you can start with the $x$ or $y$ axis as the line through origin and try to draw pictures and take the advantage of working in $\mathbb{R}^{2}$.
Since you are asked to work in $\mathbb{R}^{2}$ so geometric thinking will be of immense help. Note that you are asked to find the eigen-value of the matrix of reflection across some line "passing" through the origin not "about" the origin. You can easily see the difference by drawing some points in $xy$ plane and the reflecting them about the origin and then across some line passing through the origin. 
\Now recall that a linear transformation from some vector space to another is completely determined by its action on the basis of the domain vector space. So, in particular you can see the reflections of the standard basis of $\mathbb{R}^{2}$ across the given line and then you will be able to determine the transformation. Write the matrix of that transformation and then eigenvalue(s) can be found easily. For understanding point of view you can start with the $x$ or $y$ axis as the line through origin and try to draw pictures and take the advantage of working in $\mathbb{R}^{2}$.