The transformation matrix for polynomial differentiation.
February 3, 2023
Here I’ll show how to find the matrix of a linear transformation by doing so for the transformation corresponding to
the differentiation of a second-degree polynomial. This transformation takes the coefficients of the polynomial to be
differentiated and returns the coefficients of the derivative, which is a first-degree polynomial. We know that
and so we have
the matrix of T is 2-by-3. (The product of two matrices has as many rows as the first matrix, and as many columns as
the second matrix. Also, two matrices can be multiplied only if the first matrix has as many columns as the second one
has rows.) Thus,
for some t1 through t6. We can see that
It is clear that t1=2, t5=1, and the remaining t’s are 0. This gives us
with the transformation matrix being
As a closing note, we could just as well have T be a mapping from R3 to R3, in which case the
transformation matrix would be 3-by-3, and one of its rows would contain only zeros.