There are tons of applications of linear algebra. Perhaps the most direct ones I have found are in quantum mechanics, without which there would be no computers. Other applications include solving the gigantic algebraic systems that arise from finite element analysis (500k x 500k systems are quite commonplace) numerical solutions of pde's. For example: you want to design the landing gear for a Boeing aircraft. But you don't want to have to build and test it physically, because that's expensive. So, you model it with finite element analysis. The result is the large linear system you must solve. You use techniques of linear algebra to solve the system. Numerical analysis, indeed, is part of the intersection of math and cs (courses, for example, are often cross-listed).
You can use linear algebra to solve problem in theoretical cs like the number of walks of length n in a directed graph (turns into an eigenvalue problem). You can use linear algebra to solve problems in optimization of processor time in operating systems. The list goes on and on. Probably the only field of mathematics with more applications than linear algebra is calculus.