# Basis state
**Basis states** are set of two or more quantum states out of which all other quantum states can be built through linear combination.

A qubit has two basis states. The standard basis states are $\lvert 0\rangle$ and $\lvert 1\rangle$. This notation is called the ket-notation. It's a way to give a name to the basis states. The names inside kets have no meaning. We could have just as easily said $\lvert\text{cat}\rangle$ and $\lvert\text{dog}\rangle$. What's important is that there are exactly two distinct quantum states. These are the states we associate complex probability amplitudes to. If we had more than two states, we'd have a [qudit](https://en.wikipedia.org/wiki/Qudit). A qubit is just a special case when the number of states is $d = 2$.

We've seen that the qubit state $\lvert\psi\rangle$ is described by two complex numbers $a,b\in\mathbb C$. We might as well write it as a column vector in $\mathbb C^2$, since this will lets us take advantage of linear algebra. The space of all possible states is called a Hilbert space. In this case, Hilbert space is the space of all possible column vectors in $\mathbb C^2$.
$$ \lvert\psi\rangle = \begin{pmatrix}a\\b\end{pmatrix},\quad\lvert\psi\rangle\in\mathbb C^2$$

We can then write the column vector $\lvert\psi\rangle$ in the following form. In which case, we recognize the familiar expression $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$:

$$\lvert\psi\rangle = a\begin{pmatrix}1\\0\end{pmatrix} + b\begin{pmatrix}0\\1\end{pmatrix}$$

So, when it comes to a qubit, everything just boils down to 2-dimensional linear algebra over complex numbers. Basis states are then just special type of vectors in this space:

$$\lvert 0\rangle = \begin{pmatrix}1\\0\end{pmatrix},\quad\lvert 1\rangle = \begin{pmatrix}0\\1 \end{pmatrix}$$

In fact, in Hilbert space $\mathbb C^2$, all kets $\lvert \psi\rangle$ are column vectors, all bras $\langle \phi\lvert$ are just row vectors, and all operators $H$ are just 2x2 matrices (the elements are, of course, complex numbers). This comes from the fact a qubit only has two states and the Hilbert space $\mathbb C$ was enough to capture all possible states.

In quantum mechanics more broadly, there may be infinite number of states. In that case, states are not 2-dimensional vectors, but rather infinite dimensional. At that point, $\mathbb C^2$ is not enough to capture all possible states, so other kinds of Hilbert spaces are used.