What are the principal applications of linear algebra in theoretical physics? I am currently doing Shankar's Principle of Quantum Mechanics, and I am wondering besides the Quantum Mechanics applications that I already know it's a myriad, what are the other things inside physics I will be able to enjoy with this time and effort used to learn linear algebra?
 A: There are lots of cases outside of QM where linear algebra can be of great use.
If we have a coupled oscillator, then the equations of motion can be solved using linear algebra with ease.
In special relativity, a 'relatively' simple application of linear algebra occurs when we define 4-vectors and the Lorentz transform matrix, which tells us how co-ordinates such as space and time transform when seen in different reference frames.
Fourier series, which are useful in numerous branches of physics, such as in the heat equation, can be seen in a linear algebra light if we consider the trigonometric functions as a basis for periodic functions, with orthogonality when we integrate a product of them over a specified interval. There is a similar idea for the Legendre polynomials, which are important in electromagnetism, amongst other areas.
In optics, (which perhaps we can view as a mix of electromagnetism and QM), it can be useful to describe polarised light using a complex vector space, allowing us to represent both phase and magnitude, and matrices can be used to model the effects of polarizers and wave-plates.
