## 2. Linear operators.

Given the vector **v***(x;y)*, the product between **v** and the real number
*a* is the vector **r***(ax;ay)*.

A set of real vectors that can be multiplied by a real number is a real
vector space
(or linear space). In this context the real number *a* is said a **scalar**.

Given two vectors **v**_{1}*(x*_{1};y_{1}) and
**v**_{2}*(x*_{2};y_{2}), their
scalar (or inner or dot) product
**v**_{1}**·v**_{2} is the real number
*x*_{1}x_{2}+y_{1}y_{2}.

The scalar product between the vector **v**_{1} and the vector
**v**_{2}
can be represented in the following way

where the vector **v**_{1} is written as a row matrix and the vector
**v**_{2} as a
column matrix. The scalar product is given by the sum of the products of the corresponding
components of the two vectors.

In these pages we shall sometime use the
bra-ket notation proposed by
P.A.M. Dirac: the row matrix is
a vector `bra`

represented by `<w|`

and the column matrix is a vector
`ket`

represented by `|v>`

. So the scalar product is

The magnitude of a geometric vector is given by the square root of its square. This quantity
is generalized by the **norm** which usually is the same

In the scalar product between the vector **L**_{1}
*(L*_{1,1};L_{1,2})
and the vector **v***(x;y)* in

the vector **L**_{1} can be interpreted as an
operator applied to the
vector **v** to univocally give the real number

In order to have an univocal correspondence between a vector **v***(x;y)* and a pair
of real numbers which could be interpreted as the components of another vector
**v’***(x’;y’)*, **L**_{1} must be joined with another
operator **L**_{2}*(L*_{2,1};L_{2,2}) such that

The ordered pair **L** of the operators **L**_{1} and
**L**_{2} is represented by a *square matrix* with two rows and two columns

where the L_{i,j} are the entries (or coefficients) of the matrix.

This matrix is called the matrix of the operator L. In general the same capital letter
is used to denote an operator and its matrix.

So we have

and we can say that applying L to |v> we obtain |v'> or that L transforms |v> in
|v'>.

The application of L to |v> can be interpreted as the product of L and |v>,
so this product is a vector such that its components are given by the sum of the products
of the entries of each row and the corresponding components in the column.

L has the following properties (*a* is a scalar)

When an operator has such properties (like, for example, the
differential operator and the
integration operator it is said to be a
linear operator.

The operator I having matrix

when applied to whatever vector gives the vector itself. This operator is the
*identity operator*.