I started reading Quantum Mechanics and I'm being told that matter and light have both a particle and wave aspect to them. It is easy to understand the particle aspect of light: experiments show that light can push an electron out of it's orbit, so one imagines that light has to behave to some extent "as if" it was a particle that can collide with an electron.
Furthermore, when we say that "light" behaves as a wave I understand this as simply meaning that light can be viewed as a specific solution of Maxwell's equation of the form $f(x - c t)$ and in particular it can be expanded into a superposition of "waves" of the form $\exp(2\pi i u (x - c t))$.
However, I am confused as to what is being meant when people say that matter behaves like a wave. In particular I am confused as to the meaning of the de Broglie relation between the momentum of a particle and it's wavelength. What is the physical meaning of this wavelength associated with matter? And how is it associated to matter?
My best guess is as follows: we can conceptually view a beam of electrons of a certain energy directed in a certain direction as behaving like a "wave". This allows us to explain certain experiments in a consistent way, by maintaining the principle of superposition: for instance when two beams collide we can view the resulting beam as a superposition of the original ones. In this way coupled with the statistical interpretation of the wave, we manage to obtain physical information on the collisions of electrons having different energy levels (within the limits of precision that are enforced by the uncertainty principle).
