You should cancel "wave-particle duality" from the list. This is an old concept which has been outdated with the advent of "modern quantum theory", which has been discovered by Heisenberg in 1925.
The important concepts are states and observables. You should start with one-particle non-relativistic quantum theory, where the states can bei represented by (up to some subtleties, you should not worry about in the very beginning) complex valued functions of the position of the particle that are square integrable, i.e., you have a function \psi(\vec{x}) which is falling off at infinity such that you can set, by choice of the overall magnitude of this "wave function"
\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} |\psi(\vec{x})|^2=1.
The physical interpretation of this wave function is that |\psi(\vec{x})|^2 \mathrm{d}^3 \vec{x} is the probability to find the particle in a small volume \mathrm{d}^3 \vec{x} around the position \vec{x}, provided the particle has been prepared in a state, discribed by this wave function \psi.
This concept sounds simpler than it is, because it causes debates about the meaning of quantum theory since it's discovery nearly 90 years ago! Don't bother yourself with these more philosophical questions, before you haven't get some understanding about the theory itself. To work with the minimal interpretation is enough, where one just states that the wave function has this probabilistic meaning and nothing else.
The linear algebra comes in through the fact that you can multiply wave functions with complex numbers and add them, i.e., you can build arbitrary linear combinations from any set of square integrable functions, and you always get another square-integrable function. All these can represent states.
Further, since the wave functions are square integrable, also the scalar product,
\langle \psi_1 | \psi_2 \rangle=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \psi_1^*(\vec{x}) \psi_2(\vec{x})
exists for any two square-integrable functions \psi_1 and \psi_2. With this scalar product the vector space of square-integrable functions becomes the Hilbert space of square integrable functions, called L^2.
The next step is to understand, how to describe observables. This is not so easily explained. What comes out after some hand-waving arguments is the concept to describe observables as so-called self-adjoint linear operators. E.g., the position of the particle is described by the mutliplication of the wave function with \vec{x}, i.e.,
\hat{\vec{x}} \psi(\vec{x}):=\vec{x} \psi(\vec{x}).
Momentum is described by taking the derivatives with respect to \vec{x}, i.e.,
\hat{\vec{p}} \psi(\vec{x})=\frac{\hbar}{\mathrm{i}} \vec{\nabla} \psi(\vec{x}).
An operator \hat{A} is self-adjoint, if for any pair of wave functions \psi_1 and \psi_2, you have
\langle \psi_1 | \hat{A} \psi_2 \rangle=\langle \hat{A} \psi_1|\psi_2 \rangle.
As an exercise you should check that both \hat{\vec{x}} and \hat{\vec{p}}, defined as said above, are self-adjoint operators.
I hope this gives some hints about, which math you need to learn to understand quantum theory. It's not easy but great fun. You should take a good introductory physics book on the subject and try to learn from it. Very good books are the Feynman Lectures on Physics (three volumes, covering mechanics, electromagnetism, and quantum mechanics). As I said, it's not an easy subject, but very rewarding!
