Operators

Operators are represented by capital letters such as A, B, and C. Operators can be represented by matrices numerically, as shown in the following diagram:

The rest of bra-ket notation will be explained as the book progresses. The next section is a very advanced treatise on bras and is optional.

Bras

A bra is a linear functional. We talk about these in Chapter 5, Transforming Space with Matrices. To help jog your memory, they are a special case of linear transformation that takes in a vector and spits out a scalar:

For instance, I could define a linear functional for every vector in ℝ²:

So that:

There are many linear functionals that can be defined for a vector space. Here's another one:

The set of all linear functionals that can be defined on a vector space actually form their own vector space called...