Exercise 17

The first two items of this question are review.

• Prove that the composition of one-to-one maps is one-to-one.

• Prove that if a linear map is not one-to-one then at least one nonzero vector from the domain maps to the zero vector in the codomain.

• Verify the statement, excerpted here, that precedes Theorem 1.8.

  … if a minimal polynomial $m(x)$ for a transformation $t$ factors as $m(x)=(x-\lambda_{1})^{q_{1}}\cdots(x-\lambda_{z})^{q_{z}}$ then $m(t)=(t-\lambda_{1})^{q_{1}}\mathbin{\circ}\cdots\mathbin{\circ}(t-\lambda_{z})^{q_{z}}$ is the zero map. Since $m(t)$ sends every vector to zero, at least one of the maps $t-\lambda_{i}$ sends some nonzero vectors to zero. … That is, … at least some of the $\lambda_{i}$ are eigenvalues.