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Homework 11 - Rank, nullity, and eigenvectors

  1. Let $A$ be an $m \times n$ matrix of rank $r$. Show that there exists an invertible $n \times n$ matrix $C$ and an invertible $m \times m$ matrix $D$ with \(D^{-1} A C = \begin{pmatrix} 1 & & & & & & \\ & 1 & & & & & \\ & & \ddots & & & &\\ & & & 1 & & & \\ & & & & 0 & & \\ & & & & & \ddots & \\ & & & & & & 0 \end{pmatrix}\) where the off diagonal entries are $0$. (Hint: choose a basis $v_1,\ldots,v_d$ for $\mathbb{R}^n$ where $v_{r+1},\ldots,v_m$ is a basis of $\mathcal Z(A)$. Then, use the basis for $\mathbb{R}^m$ of the form $Av_1,\ldots,Av_r,w_{r+1},\ldots,w_m$.)

  2. What is the rank and nullity of \(\frac{d^i}{dx^i}: \operatorname{Poly}_n(\mathbb{R}) \to \operatorname{Poly}_n(\mathbb{R})\) for each $i > 0$?

  3. If $v$ is an eigenvector with eigenvalue $\lambda$ for an invertible $T: V \to V$, show that $v$ is also an eigenvector for $T^{-1}$. What is its eigenvalue?

  4. Let $T,S: V \to V$ be linear transformations. Show that if $v$ is an eigenvector for both $T$ and $S$, then $v$ is an eigenvector for $T \circ S$. Find a formula for the eigenvalue of $v$ for the composition $T \circ S$ in terms of the eigenvalues for $T$ and $S$.

  5. Suppose $T,S: V \to V$ are linear transformations with $T \circ S = S \circ T$. Show that \(S: E_\lambda(T) \to E_\lambda(T)\) is well-defined. (Hint: you need to show that if $v$ is a $T$-eigenvector with eigenvalue $\lambda$ then $S(v)$ is also a $T$-eigenvector with eigenvlaue $\lambda$.)