Let me elaborate on my comment.
If you're looking for some "linear algebra" treatment of PDEs you will get into the field called functional analysis. Functional analysis is some kind of infinite-dimensional linear algebra. Here we are working with function spaces for example the space of square integrable functions $L^2$ ($f$ is in $L^2$ if $\int |f|^2 < \infty$). It can be quickly seen that there will be no finite basis that spans the complete space, so our space is infinite dimensional.
So, we are working with function spaces, in PDE we are looking for function spaces where our solutions of the PDEs live. That is, we are finding functions that satisfy the equation. It turns out that looking at the derivative in a classical sense doesn't give you much tools to study certain properties of the solutions like regularity (how "differentiable" your function is) and so on. So we interprete the derivative in a "weak" (or distributional) sense. In this way we have a larger class of functions that satisfies our equation and we can apply the tools of functional analysis to that. These spaces are called the Sobolev spaces. If you would like me to explain more in detail why we would study those spaces, do ask.
This was actually a short summary why we would need functional analysis. Another thing you will need is measure theory. Measure and integration theory studies a different type of integral than the one you're used to (the Riemann integral) namely the Lebesgue integral. This integral has many more nice properties for example you have nice theorems that state that under mild conditions on $f_n, f$ that
$$\int f_n \to \int f.$$
The Riemann integral also possesses this property, but under quite "unnatural" conditions (uniform convergence for example).
I'm not sure how far you exactly are but if I understood correctly, you're having the engineering mathematics knowledge.
So, you would need to know:
Basic Real Analysis:
- The blog by Terence Tao: http://terrytao.wordpress.org/ contains nice lecture notes. You're looking for something like:
- Bartle - Real Analysis. This book explains basic real analysis. Here you will get the formal definition of continuity, convergence, the Riemann integral and so on. This is really a prerequisite for measure theory.
- Pugh - Real Mathematical Analysis is another option as is
- Rudin - Principles of Mathematical Analysis. This books quite hard to study the subject from.
- Schilling - Measure, Integrals and Martingales. This is a quite inexpensive book and all the solutions are available online. It is a very gentle introduction. You will only need the measure theory bit, not the probability bit (with the martingales)
- Folland - Real Analysis. This is my favorite, it is also not very easy to learn the subject from but worth the effort. And last but not least:
- Rudin - Real and Complex Analysis. I suppose the first few chapters are sufficient, but here the same holds as for the other book by Rudin which I have mentioned.
- Terence Tao's blog. This website also has some nice (free!) notes.
- Werner - Funktionalanalysis. If you can read German, this is a gentle introduction. You only need to know the Banach spaces and Hilbert spaces, for the moment you will not need the topological vector spaces bit.
- Conway - A course in functional analysis. This is also quite dense, but might be worth the effort. You only need to know the basic theorems, but these books use a "measure theoretic" approach.
- Rynne and Youngson - Linear Functional Analysis. This is an undergraduate book, and I think it contains all you need.
Partial Differential Equations:
- Evans - Partial Differential Equations. This is a standard book for such courses, I have studied the subject as well from this book.
- Krylov - Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Is also nice if you prefer a more functional analytic approach (this books turns around the study of Sobolev spaces on domains or on the whole space).
This was some short list of suggestions. If you have any questions do ask!