The * Dirac bra-ket notation* is a concise way to represent objects in a complex vector space .

A ** ket**, denoted by , is a vector . Since a linear operator maps a vector to another vector, we have .

A ** bra**, denoted by , is often associated with a ket in the form of an

*, denoted by . If a ket is expressed as a column vector, the corresponding bra is the conjugate transpose of its ket, i.e. . The inner product can therefore be written as the following matrix multiplication:*

**inner product**or in the case of functions:

Since a linear operator acting on a ket is another ket, we can express an inner product as:

where .

If , is the expectation value (or average value) of the operator .

As mentioned above, bras and kets can be represented by matrices. Therefore, the multiplication of a bra and a ket that involves a linear operator is associative, e.g.:

You can verify the above examples using a 2×2 matrix with complex elements to represent the operator acting on a vector in . The three examples reveal that:

- produces another ket.
- results in another bra. This is because , where is a scalar; and if , the only possible identity of is a bra.
- , which is called an
, is an operator because , i.e. maps the ket to another ket . In other words, the operator transforms the vector in the direction of the vector , i.e. projects onto .*outer product* - The product of two linear operators is another linear operator: .