Linear Operator

linear-algebra

Definition

Linear Operator

Let $V$ be a vector space. A linear operator $T:V\to V$ is an operator and a linear map, i.e.,

1. it maps from a vector space $V$ back to itself $V$, and
2. it is linear.

Bra-Ket Notation

In bra-ket notation, a linear operator $T$ acts as

$$⟨x|T|y⟩=⟨x,Ty⟩.$$

Inference from Basis State

If you know what a linear operator does to basis vectors, then you know what it does to every vector.

Example

Let $X:{\mathbb{C}}^2\to {\mathbb{C}}^2$ be a linear operator. Suppose

$$X|0⟩=|1⟩\text{and}X|1⟩=|0⟩.$$

Then for

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩,$$

we get:

$$\begin{array}{rl}X|\psi ⟩& =X(\alpha |0⟩+\beta |1⟩)\\ & =X(\alpha |0⟩)+X(\beta |1⟩)\\ & =\alpha X|0⟩+\beta X|1⟩\\ & =\alpha |1⟩+\beta |0⟩\end{array}$$

Hence, we can infer that $X$ swaps the coefficients.