### Linear Transformation

##### 2. Operations of linear transformations and Hom space

The operations for matrices/linear transformations between euclidean spaces may be generalized.

The addition of two linear transformations `T`, `S`: `V` → `W` is given by

(**T** + **S**)(`u`) = **T**(`u`) + **S**(`u`).

The scalar multiplication of a number `c` to a linear transformation `T`: `V` → `W` is given by

(`c`**T**)(**u**) = `c`**T**(`u`).

The composition of linear transformations `T`: `V` → `W` and `S`: `U` → `V` is given by

(**TS**)(`u`) = **T**(**S**(`u`)).

The verification of the linearity of **T** + **S**, `c`**T**,
`TS` can be found in this exercise. The generalization of the transpose will be the dual transformation.

Example The transformation `T`(`f`) = `f`'' - 4`f`' + 3`f`: `C`^{2}(**R**) → `C`(**R**) is linear because

**T** = `DD` + 4`D` - 3`id`

is a linear combination of transformations `DD`, `D`, `id`.
The identity `id` and the derivative `D` are known to be linear.
As the composition of two linear transformations, `DD` is also linear.
Note that since `T` is linear, the differential equation `f`'' - 4`f`' + 3`f` = 3`t` - 1 is a linear equation.

Example Fixing an `n` by `n` matrix `A`,
matrices `X` commuting with `A` are the solutions of the equation `AX` = `XA`.
The equation is linear because it can be written as `AX` - `XA` = `O`.
The left side `AX` - `XA` is a linear transformation of `X` because
it is a linear combination of `X` → `AX` and `X` → `XA`, which are known to be linear.

Denote by `Hom`(`V`, `W`) the collection of all linear transformations from `V` to `W`. It is easy to verify that the two operations satisfy the eight properties in the definition of vector spaces. For example, for `T`, `S` ∈ `Hom`(`V`, `W`) and `u` ∈ `V`, we have

(`T` + `S`)(`u`)

= `T`(`u`) + `S`(`u`) (definition of `T` + `S`)

= `S`(`u`) + `T`(`u`) (`w` + `w'` = `w'` + `w` in `W`)

= (`S` + `T`)(`u`). (definition of `T` + `S`)

This verifies the first property. The other properties can be similarly verified. Therefore `Hom`(`V`, `W`) is a vector space.

In the special case `V` = **R**^{n}, `W` = **R**^{m}, `Hom`(**R**^{n}, **R**^{m}) can be naturally identified with the collection `M`(`m`, `n`) of `m` by `n` matrixes. Moreover, it is easy to see that the addition and scalar multiplication in `Hom`(**R**^{n}, **R**^{m}) and `M`(`m`, `n`) match in the identification.