To me, representation theory means a lot more than abstract linear algebra. Of course, one can define terms however one wishes, including in such a way that these two become synonymous—and, also of course, maybe you were intentionally using the terms informally to give an idea of your meaning without going into too much technicality—but, to me, abstract linear algebra is the study (for each field $k$) of the category whose objects are $k$-vector spaces and whose morphisms are $k$-linear maps, whereas representation theory is the study of the category whose objects are triples $(G, X, \alpha)$ of a group $G$, a space $X$, and an action $\alpha$ of $G$ on $X$ (let's stipulate—though not everyone restricts representation theory to this setting!—that $X$ is a $k$-vector space, and $\alpha$ is a linear action), with morphisms being intertwiners in an obvious sense.
That is, in some sense, I view representation theory as somehow 'reifying' the morphisms of abstract linear algebra to become objects (a perspective which could, and sometimes does, lead one down the path of $n$-categories).