950 Extra Problems Modules

Modules and Linear Algebra

  • Prove the Cayley-Hamilton theorem.
  • Prove that the minimal polynomial divides the characteristic polynomial.
  • Prove that the cokernel of \(A\in \operatorname{Mat}(n\times n, {\mathbf{Z}})\) is finite \(\iff \operatorname{det}A \neq 0\), and show that in this case \({\left\lvert {\operatorname{coker}(A)} \right\rvert} = {\left\lvert {\operatorname{det}(A)} \right\rvert}\).
  • Show that a nilpotent operator is diagonalizable.
  • Show that if \(A,B\) are diagonalizable and \([A, B] = 0\) then \(A,B\) are simultaneously diagonalizable.
  • Does diagonalizable imply invertible? The converse?
  • Does diagonalizable imply distinct eigenvalues?
  • Show that if a matrix is diagonalizable, its minimal polynomial is squarefree.
  • Show that a matrix representing a linear map \(T:V\to V\) is diagonalizable iff \(V\) is a direct sum of eigenspaces \(V = \bigoplus_i \ker(T -\lambda_i I)\).
  • Show that if \(\left\{{\mathbf{v}_i}\right\}\) is a basis for \(V\) where \(\dim(V) = n\) and \(T(\mathbf{v}_i) = \mathbf{v}_{i+1 \operatorname{mod}n}\) then \(T\) is diagonalizable with minimal polynomial \(x^n-1\).
  • Show that if the minimal polynomial of a linear map \(T\) is irreducible, then every \(T{\hbox{-}}\)invariant subspace has a \(T{\hbox{-}}\)invariant complement.