I am considering taking a course in linear algebra and wanted to get a survey to see how important it is. My goal is not just to understand how to push the buttons in some software (e.g. GW Vistas, Aquiferwin32), but to really understand how groundwater modeling works.
Relatively important, depending on the basic principles of the modeling you are interested in, and to what extent you want to get yourself involved in it.
In many areas of computational geophysics, e.g. atmospheric, oceanic, hydrological modeling etc., there are modelers, modelers, and modelers. First group of modelers get model output from somebody else, do some analysis, make some plots, and write about what they found. Then, there are modelers who design their own experiments, play with tunable parameters, maybe even change a few lines in the code and run the model. Finally, there are modelers - these spend most of their time on building and improving models, both in terms of numerical methods and software design. They could go for few years without publishing a single paper, but boy did they work hard.
If you are interested in computational science aspects of modeling, i.e. the inner workings of the model and how to improve it, understanding linear algebra as well as numerical methods will prove to be very valuable. While it is likely that you will never end up having to implement that matrix inversion or sparse matrix multiply yourself, and instead use one of many well documented production libraries, it is important to understand how they do it and why they do it in a particular way. From my own experience, I can say that most of the time it won't matter, but sooner or later there comes a situation where having knowledge about how low-level operations work will save you from pulling your hair out for weeks.
And this does not go only for linear algebra - I would extend this advice and urge you to strive at also understanding differential equations, discretization, computer architecture, floating-point arithmetic.
These days, not so much.
Even people who write their own sophisticated mesh based groundwater modeling codes usually make extensive use of sparse/dense numerical libraries/solvers that are written by linear algebra experts (e.g., UMFPACK/MUMPS/PETSc etc.). The author of these libraries really know their stuff.
So if you are writing your own groundwater modeling code you simply call the above libraries to solve your equations.
Off course 20 years ago when there were no such libraries people had no choice. That is why codes written in 80s and early 90s typically have their own (often poor performing) solver.
Edit: If you're just using a code then don't bother with linear algebra at all. The only thing you need to know is that for large problems you need to use an 'iterative' solver.