Let P₂ be the vector space of polynomials of degree at most Let T: P₂ → P₂ by T (p(x)) = p(x) + x p(x). a) Show that T is a linear transformation b) Determine the kernal of T c) Show that T is an isomorphism d) Determine T-¹

Show step by step solution

3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.

4. Let V be a vector space. Prove that a) The zero transformation T(v) = b) The identity transformation T(v) = v for all v EV is a linear transformation. = 0 for all v EV is a linear transformation. CS Scanned with CamScanner

Please show work