To find the matrix of a given linear transformation, L, from V to V, given an ordered basis for V, apply L to each of the basis vectors in turn, and write the result as a linear combination of the basis vectors. The coefficients of that linear combination form one column of the matrix. Here, the second basis "vector" is the function "x". Applying the given linear transformation to that gives because the first derivative is 1 and the second derivative is 0. The second column of the matrix representation is
Do the same thing, applying L to 1, , , to find the other three columns.
To find the matrices and do the same thing with the identity transformation. For example, the identity map takes the second basis vector in [tex]\beta[tex], "x", to x which we need to write as a linear combination in the basis [tex]\gamma[tex]: so the second column of is
Do the same thing, the other way around, to find the matrix or find the inverse of that function.
To find the matrix , you can either
1) Do the same as in (a): Applying the linear transformation to 1+ x gives so the second column of is
2) Multiply: which converts from basis to basis , applies the linear transformation in that basis, then converts back.
Of course, applying the linear transformation "n" times is the same as multiplying by the corresponding matrix n times. And because is diagonal, it is easy to take powers: