I was watching a video on "How Does a Quantum Computer Work?".
I'm confused about what they mean by: "Although the qubits can exist in any combination of states, when they are measured they must fall into one of the basis states."
From what I know about linear algebra, if we represent the state of a qubit by $|\psi\rangle$ it can be written like $\alpha|x\rangle + \beta |y\rangle$ (where $|x\rangle$ and $|y\rangle$ form a basis) or $\gamma (|x\rangle+|y\rangle) + \delta (|x\rangle-|y\rangle)$ or even $A(|x\rangle+|100y\rangle) + B |y\rangle$! What I mean is that no set of basis states is unique.
So, in reality which set of basis states can a qubit (or more generally a quantum system of qubits) actually collapse to? Can an actual measurement land us with a basis state like $(|0\rangle + |1\rangle)$ or $(|0\rangle - |1\rangle)$ ? Or is only $|0\rangle$ and $|1\rangle$ possible? Also does the basis vector which a qubit can land up in have to have norm $1$ (i.e. must it be an element of an orthonormal basis)?