Like you, I also had little motivation for Linear Algebra at university, especially as a young engineering student who could care less about abstract proofs. However, I've come to both appreciate and really enjoy linear algebra once I saw how it ties together so many other subjects and also allows one to develop nice algorithms for numerical computations.
My advice is to stay the course, and focus on the unifying nature of Linear Algebra. To wit:
-The formula of the Fourier Coefficients is not intuitive based on the pure calculus derivation. However, linear algebra will let you see that each coefficient is really the inner product of a "basis function" (like a basis vector) with the function you are approximating. Thus, your function acts like a vector, and the various sine and cosine functions are the "elementary vectors" that form the basis of the trigonomentric functon space.
If you can formulate a problem in terms of array and vector operations, then there usually exists an extremely fast algorithm for solving it. This is especially apparent in interpreted languages like R and Python, where a lazy "for" loop will add an order of magnitude to your running time.
Eigenvector decomposition of covariance matrices is the heart of a lot of statistical and machine learning approaches, which use eigenvectors to find lower-dimensional representations of a dataset.
Quantum mechanics' formalism is based on linear algebra and Fourier series, so you will have access to this area of science.
And many, many more uses. Really, Linear Algebra has turned out to be an incredibly unifying and intellectually broadening subject for me, to the extent that I actually got interested in Abstract Algebra (once you go abstract you don't go back ;-P