Dirac notation for quantum states

How to read the bra–ket notation?

Quantum states are usually described using bra–ket notation, also known as Dirac notation.

In Dirac notation, a ket $∣ v ⟩$ represents a state of a quantum system and is a vector in a complex vector space $C^{n}$ . A bra $⟨ f ∣$ denotes a linear map $f : C^{n} \to C$ , and the application of $⟨ f ∣$ to $∣ v ⟩$ is written as $⟨ f ∣ v ⟩$ .

The ket $∣ v ⟩$ can be represented as a column vector:

∣ v ⟩ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ v_{1} v_{2} ⋮ v_{n} ⎦ ⎥ ⎥ ⎥ ⎥ ⎤,

and the bra $⟨ f ∣$ as a row vector:

⟨ f ∣ = [f_{1} f_{2} \dots f_{n}] .

In this representation, a bra next to a ket simply denotes matrix multiplication of a row vector with a column vector:

⟨ f ∣ v ⟩ = [f_{1} f_{2} \dots f_{n}] ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ v_{1} v_{2} ⋮ v_{n} ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ = i = 1 \sum n f_{i} v_{i} .