Ket (State)

Let $\left\lvert x \right\rangle$ (ket x) be a $N \times 1$ matrix where $N$ is any positive integer (number with no decimals and is greater than 0).

Just like how $x$ was a variable used to represent ONE number, $\left\lvert x \right\rangle$ can be used to represent any $N\times 1$ matrix.

Example:

\begin{gather*} \left\lvert x \right\rangle=\begin{pmatrix} 1 \\ 2 \\ 7 \end{pmatrix}\quad \left\lvert \psi \right\rangle=\begin{pmatrix} 9 \\ 9 \\ 0 \\ 0 \\ 0 \end{pmatrix}\quad\left\lvert \phi \right\rangle=\begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{gather*}

Note that $\psi$ (psi) is pronounced sye and $\phi$ (phi) is pronounced fye. They are just like $x$ but idk why quantum physicists love greek characters.