Consider the vector space p2 of polynomials of degree at most 2. let t be the linear transformation given by t(p(x)) = p(x − 1) + p(x + 1). find the matrix of t with regard to the basis {1, x, x 2 }.

Observe what happens to the basis components under the transformation :





So we can represent with the matrix



To check this If we take an arbitrary polynomial , i.e. the vector



corresponding to the polynomial , we get



Substituting into the rule above, we get




as expected.