Homework #3

1. Let \( f:\mathbb{R}^2 \to \mathbb{R} \) be defined by \( f((a,b))=a \) and \( g:P_2 \to P_2 \) be defined by \( g(at^2+bt+c)=2at+b \) (differentiation).
   [Recall that \(P_2 = \{ at^2+bt+c \;|\; a,b,c \in \mathbb{R} \} \)]
   We can define a new linear map \( f \otimes g : \mathbb{R}^2 \otimes P_2 \to \mathbb{R} \otimes P_2 \) by letting \( (f \otimes g)\Big((a,b) \otimes (ct^2+dt+e)\Big) = f((a,b)) \otimes g(ct^2+dt+e) \) and extending linearly (recall that not all tensors are pure tensors).
   
   Let \( \alpha = \{ 1 \} \), \( \beta = \{ {\bf i}, {\bf j} \} \), and \( \gamma = \{ t^2,t,1 \} \) be the standard bases for \(\mathbb{R}\), \(\mathbb{R}^2\), and \(P_2\) respectively.
   
   First, explicitly list the elements in the bases \( \alpha \otimes \gamma \) and \( \beta \otimes \gamma \) and state the dimensions of \(\mathbb{R}^2 \otimes P_2 \) and \( \mathbb{R} \otimes P_2 \).
   
   Then find \( [f]_{\beta}^{\alpha} \), \( [g]_{\gamma}^{\gamma} \), and \( [f \otimes g]_{\beta \otimes \gamma}^{\alpha \otimes \gamma} \).
2. Let \( {\bf a} = (1,2), {\bf b} = (3,4), {\bf c} = (5,6) \), and \( {\bf d}=(1,-1) \). Recall that \( T(\mathbb{R}^2) = \mathbb{R} \oplus \mathbb{R}^2 \oplus \left(\mathbb{R}^2 \otimes \mathbb{R}^2\right) \oplus \cdots \) is the tensor algebra generated by the vector space \( \mathbb{R}^2 \). Also, recall that the standard basis \( \{ {\bf i}, {\bf j} \} \) for \( \mathbb{R}^2 \) generates a standard basis for \( T(\mathbb{R}^2) \). Specifically, \( \{ 1,{\bf i}, {\bf j}, {\bf i}\otimes {\bf i}, {\bf i}\otimes {\bf j}, {\bf j}\otimes {\bf i}, {\bf j}\otimes {\bf j}, {\bf i}\otimes {\bf i} \otimes {\bf i}, \dots \} \)
   
   
   1. Expand \( \left(2-{\bf a} \otimes {\bf b}\right) \otimes \left(-2+3{\bf d}+{\bf c}\otimes {\bf b}\right) \) in terms of the standard basis for \( T(\mathbb{R}^2) \).
   2. Recall that by quotienting a tensor algebra \( T(V) \) by the ideal generated by elements of the form \( X \otimes Y - Y \otimes X \), we got the Symmetric Algebra \( S(V) \). When working in the symmetric algebra we suppressed the tensor symbols (writing \( {\bf a}{\bf b} \) instead of \( {\bf a}\otimes {\bf b} \)).
      
      Simplify your result from part (a), assuming you are working in the symmetric algebra \( S(\mathbb{R}^2) \). Also, (up to isomorphism) what exactly is \( S(\mathbb{R}^2) \)? (This part of your answer should expressed using terms a calculus student would understand.)
      
      
   3. Recall that by quotienting a tensor algebra \( T(V) \) by the ideal generated by elements of the form \( X \otimes X \), we got the Exterior Algebra \( \Lambda(V) \). Also, recall that when working in the exterior algebra we use wedges to denote multiplication instead of the tensor symbols (i.e. we write \( {\bf a}\wedge{\bf b} \) instead of \( {\bf a}\otimes {\bf b} \)).
      
      Simplify your result from part (a), assuming you are working in the exterior algebra \( \Lambda(\mathbb{R}^2) \).
      
      
3. Let \( T : V \to V \) be a linear endomorphism.
   1. Suppose that \( \lambda,\mu\), and \( \sigma \) are distinct eigenvalues of \( T \). First, show that if \( U = E_\lambda + E_\mu \), then \( U = E_\lambda \oplus E_\mu \). Then consider \( W = E_\lambda + E_\mu + E_\sigma \) (the sum the 3 corresponding eigenspaces). Show \( W = E_\lambda \oplus E_\mu \oplus E_\sigma \).
      [In both cases, you are being asked to show that these sums are direct.]
   2. Suppose that some subspace \( W \) of \( V \) is \( T \)-invariant. Show that \( W \) is \( f(T) \)-invariant for all \( f(t) \in \mathbb{F}[t] \).
   3. Let \( {\bf v} \in V \) be some non-zero vector and define \( W =\mathrm{span}\{ {\bf v}, T({\bf v}), T^2({\bf v}), \dots \} \). Show that \( W \) is \( T \)-invariant. Then consider \( S = T|_W \) (the restriction of \( T \) to \( W \)). Suppose that \( t^3-2t+5 \) is the minimal polynomial for \(S \). Show that \( \beta = \{ {\bf v}, T({\bf v}), T^2({\bf v}) \} \) is a basis for \( W \). Also, find \( [S]_\beta \).
   
   
   
4. Find the Jordan form of \( A = \begin{bmatrix} 2 & -1 & -1 & -1 & 0 & -1/2 & -1 \\ 2 & 4 & -1 & 3 & -1 & -1/2 & -1 \\ -6 & -4 & 5 & -5 & 1 & 1 & 3 \\ -4 & -3 & 3 & -3 & 2 & 2 & 3\\ -6 & -4 & 3 & -5 & 3 & 3/2 & 3 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 8 & 5 & -5 & 7 & -2 & -5/2 & -3 \end{bmatrix} \) as well as an invertible matrix \( P \) such that \( P^{-1}AP \) is the Jordan form of \( A\). You may use Maple to do your computations, but I want to see your work building up a basis of generalized eigenvectors (i.e. don't just use the "JordanForm" command).
   
   Copy/Paste into Maple:

   A := Matrix(7, 7, {(1, 1) = 2, (1, 2) = -1, (1, 3) = -1, (1, 4) = -1, (1, 5) = 0, (1, 6) = -1/2, (1, 7) = -1, 
                      (2, 1) = 2, (2, 2) = 4, (2, 3) = -1, (2, 4) = 3, (2, 5) = -1, (2, 6) = -1/2, (2, 7) = -1,
                      (3, 1) = -6, (3, 2) = -4, (3, 3) = 5, (3, 4) = -5, (3, 5) = 1, (3, 6) = 1, (3, 7) = 3, 
                      (4, 1) = -4, (4, 2) = -3, (4, 3) = 3, (4, 4) = -3, (4, 5) = 2, (4, 6) = 2, (4, 7) = 3, 
                      (5, 1) = -6, (5, 2) = -4, (5, 3) = 3, (5, 4) = -5, (5, 5) = 3, (5, 6) = 3/2, (5, 7) = 3, 
                      (6, 1) = 0, (6, 2) = 0, (6, 3) = 0, (6, 4) = 0, (6, 5) = 0, (6, 6) = 2, (6, 7) = 0, 
                      (7, 1) = 8, (7, 2) = 5, (7, 3) = -5, (7, 4) = 7, (7, 5) = -2, (7, 6) = -5/2, (7, 7) = -3});