Homework 12: Compositions, Inverse Transformations, Isomorphisms

Instructions. Problems 7 and 8 are optional.

1. Section 8.2, #20, 26.
2. Section 8.3, #2, 4, 10(b). (Please provide justifications.)
3. Consider \(T:P_2\to P_2\) defined by \(T(p(x))=p'(x)\), where \(p'(x)\) is the derivative of \(p(x)\). Prove that \(T\) is linear. Then determine \(\ker(T)\) and \(R(T)\). Verify that Theorem 2.5 holds for \(T\).
4. Consider the linear transformation \(T: P_3\to P_3\) given by \(T(p(x))=xp'(x)\). Determine \(\ker(T)\) and \(R(T)\).
5. Consider \(T:P_2\to P_2\) given by \(T(p(x))=p(x)+p'(x)\).
   1. Prove that \(T\) is linear.
   2. Prove that \(T\) is an isomorphism.
   3. Give \(\ker(T)\) and \(R(T)\).
6. Give an example of a linear operator \(T:\bR^2\to \bR^2\) such that \(\ker (T)=R(T)\).
7. (Optional) Prove that if \(T:V\to W\) is an isomorphism, then so is \(T^{-1}:W\to V\).
8. (Optional) Let \(T:V\to W\) be a linear transformation, and let \(Y\) be a subspace of \(W\). The inverse image of \(Y\), denoted \(T^{-1}(Y)\), is defined to be $$ T^{-1}(Y)=\{\bv\in V: T(\bv)\in Y\} $$ Prove that \(T^{-1}(Y)\) is a subspace of \(V\).
   Note: The symbol \(T^{-1}\) is used to represent both the inverse image, as defined in this problem, and the inverse transformation, as defined in class on Wednesday and used in Problem 6. Although the same symbol is used for both, they are different concepts.