David Robertson ______________________________________________________ djr0045

Week 6 _____________________________________________________________ Problem 4.4.13

http://calclab.math.tamu.edu/~djr0045/p4-4-13.htm

PROBLEM:

Find the matrix (with respect to standard bases) of the linear operator

defined by

PROBLEM STATED IN MY OWN WORDS:

Consider the linear function

defined by

What is the matrix representing L with respect to the traditional bases?

SOLUTION:

We know that P₂(t) = {t²,t,1} and that P₁(t) = {t,1}, which mean, respectively, a polynomial P of degree 2 with respect to t and a polynomial P of degree 1 with respect to t. Incidentally, when doing an integration one needs to know that P₂(tau) = {tau^2,tau,1} and P₁(tau) = {tau,1}.

The following are necessary calculations with respect to P₁.

The matrix L is formed in the following format.

Thus, the above matrix is L. The answer can be stated more formally as follows.