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
(cT)(u) = cT(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, cT, TS can be found in this exercise. The generalization of the transpose will be the dual transformation.
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 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 = 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.