### 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: VW is given by

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

The scalar multiplication of a number c to a linear transformation T: VW is given by

(cT)(u) = cT(u).

The composition of linear transformations T: VW and S: UV is given by

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

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

Example The transformation T(f) = f'' - 4f' + 3f: C2(R) → C(R) is linear because

T = DD + 4D - 3id

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'' - 4f' + 3f = 3t - 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 XAX and XXA, 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, SHom(V, W) and uV, 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 = Rn, W = Rm, Hom(Rn, Rm) 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(Rn, Rm) and M(m, n) match in the identification.