Earlier, we talked about linear algebra from a concrete perspective, as the study of large groups of numbers and operations. We will now study linear algebra from a more abstract perspective, which is useful for when we get to wave mechanics.
The fundamental objects of linear algebra are really vector spaces. Normally, we denote vectors by making them bold, or adding an arrow, e.g. $\mathbf v$ or $\vec v$. Now we will shift our notation: we will write vectors as $|\psi\rangle$ and rename them kets.
We still have the dagger operation from linear algebra. When we dagger a column vector, we get a row vector. Analogously, when we dagger a ket, we get a bra. We let $|\psi\rangle^\dagger=\langle\psi|$. Note that we use the same letter to denote a ket and its dagger, except we have different brackets surrounding it.
Given a row and column vector, we can multiply them to obtain a number, their inner product. We multiply a bra and a ket to get a number, their inner product denoted $\langle\phi|\psi\rangle$. Note one of the bars has disappeared. The inner product of a ket and its dagger, $\langle\psi|\psi\rangle$, is the length of the ket.
The most important principle of quantum mechanics is:
In the next part, we will examine the analogs of matrices.