Question

A) Define (t ⋅ p^2 ⋅ p^2) by (t(p) = p(1) + p(1/t) + p(1/t^2)). Find (t(p)) when (p(t) = 1 + t + t^2). Is (p) an eigenvector of (t)? If so, what is its eigenvalue?

b) Find (t(p)) when (p(t) = 2 + t). Is (p) an eigenvector of (t)? If so, what is its eigenvalue?

Answer 1

The question asks to find the value of a linear transformation applied to certain polynomials and to determine if these polynomials are eigenvectors of the transformation, along with their eigenvalues.

Step-by-step explanation:

The student's question involves finding the value of a linear transformation (t) applied to a polynomial (p) and determining whether (p) is an eigenvector of (t). The linear transformation is defined by t(p) = p(1) + p(1/t) + p(1/t^2). For the polynomial p(t) = 1 + t + t^2, we substitute t with 1, 1/t, and 1/t² respectively into p(t) and add the results to find t(p).

An eigenvector (p) of a linear transformation (t) satisfies the equation t(p) = λp, where λ is the eigenvalue. Once t(p) is found, we can check whether (p) satisfies this condition with any λ and, if so, determine the eigenvalue. The process will be repeated for p(t) = 2 + t to find another potential eigenvector and its eigenvalue.

Unfortunately, the provided reference information does not directly assist with solving the question. Therefore, we must apply the definitions and properties of linear transformations and eigenvectors to solve the problem independently.

We also remind students that the quadratic formula, mentioned in one of the unrelated reference materials, is used to solve quadratic equations of the form ax^2 + bx + c = 0.