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Due Wednesday March 22, in class

• Section 3.3 Inverses
  • Read Section 3.3. Do the True/False questions (pp. 191-192). Do Problem 3.68. WeBWorK reading question due March 3 @ 9:50 am.
  • In class (group work): 3.69(l), 3.70(l), 3.78. Final due date: March 22
  • Homework: 3.69(c)(f)(k)(m), 3.70(f), 3.73, 3.79, 3.82, 3.85, 3.89, 3.90 (use proof by induction for (b)), 3.96
  • Hand In: 3.69(k)*, 3.79*, 3.82*, 3.90* (use proof by induction for (b))
  • 3.69(k)*. Complete Problem 3.69(k) on Page 192, but with the matrix \[\left[\matrix{1&0&2&0\cr 0&1&0&0\cr -1&0&1&1\cr 1&1&1&1\cr}\right]\]
  • 3.79*. Complete Problem 3.79 on Page 194, but where \[B=\left[\matrix{1&2&3\cr 1&-1&1\cr 2&1&3\cr}\right]\ \mbox{and}\ C=\left[\matrix{2&1&3\cr 0&0&0\cr 1&0&-1\cr}\right] \]
  • 3.82*. Complete Problem 3.82 on Page 195, but with the matrix \(A\) satisfying the equation \(A^3-3A^2+6A-I={\bf 0}\).
  • 3.90*. Similar to Problem 3.90 on Page 195: Let \(P\) and \(D\) be \(n\times n\) matrices with \(P\) invertible. (a) Prove that \((PDP^{-1})^2=PD^2P^{-1}\); (b) What is the corresponding formula for \((PDP^{-1})^3\)? for \((PDP^{-1})^n\)? Prove your answer (use a proof by induction for the second part of (b)).
  • WeBWorK assignment for Section 3.3 due March 20 @ 11 pm
• Section 3.4 The LU Decomposition
  • Read Section 3.4. Do Problems 3.104(a)(b), 3.105(a)(d). WeBWorK reading question due March 6 @ 9:50 am.
  • In class (group work): 3.104(c), 3.105(c), 3.110. Final due date: March 24
  • Homework: 3.105(d) (do NOT use any row interchanges or row multiples!), 3.108, 3.111, 3.112, 3.113, 3.114, 3.115
  • Hand In: 3.105(d)* (do NOT use any row interchanges or row multiples!), 3.113*
  • 3.105(d)*. Complete Problem 3.105(d) on Page 214, but with the matrix \[\left[\matrix{2&3&2\cr 1&2&3\cr 1&3&1\cr}\right]\]
  • 3.113*. Complete Problem 3.113 on Page 215, but with \(R_1\) and \(R_4\) exchanged and \(R_2\) and \(R_3\) exchanged.
  • WeBWorK assignment for Section 3.4 due March 22 @ 11 pm
• Section 3.5 The Matrix of a Linear Transformation
  • Read Section 3.5. Do the True/False questions (p. 230). Do Problems 3.121(b), 3.123(b). WeBWorK reading question due March 10 @ 9:50 am.
  • In class (group work): 3.121(a), 3.123(b), 3.124(a) (actually check that M is given by the indicated formula). Final due date: March 31
  • Homework: 3.118, 3.120, 3.121(d), 3.122(a), 3.124 (actually check that M is given by the indicated formula), 3.125, 3.128 (only use the matrix A in those parts that refer to A), 3.130, 3.136
  • Hand In: 3.121(d)*, 3.125(c)*, 3.128(h)* (note part (h) does not make use of the matrix A), 3.136*
  • 3.121(d)*. Complete Problem 3.121(d) on Page 232, but with the basis \(\mathcal{B}=\{[1,3,1]^t, [-1,1,-2]^t,[2,1,-3]^t\}\).
  • 3.125(c)*. Complete Problem 3.125(c) on Page 233, but with the matrix \[A=\left[\matrix{3&2&1\cr 5&6&4\cr}\right]\] and the basis \(\mathcal{B}=\{[1,1,2]^t, [1,-2,0]^t,[1,-2,-1]^t\}\).
  • 3.128(h)*. Complete Problem 3.128(h) on Page 234, but with the linear transformation \(L\ :\ M(4,2)\to M(2,4)\), where \(L(X)=X^t\).
  • 3.136*. Similar to Problem 3.136 on Page 235: Let \(\mathcal{B}=\{X_1,X_2,\ldots,X_k\}\) be a subset of a vector space \(\mathcal{V}\) and let \(L\ :\ \mathcal{V}\to \mathcal{W}\) be an isomorphism of vector spaces. Prove that if \(\mathcal{B}\) is linearly independent in \(\mathcal{V}\) then \(\{L(X_1),L(X_2),\ldots,L(X_k)\}\) is linearly independent in \(\mathcal{W}\).
  • WeBWorK assignment for Section 3.5 due March 27 @ 11 pm
• Extra Credit Problem (5 extra points available), credit still available if not already gained on Homework 3.
  • Turn in with Homework 4 (for the extra credit): Let \(U\) and \(W\) be subspaces of the vector space \(V\). Show that \(U\cup W\) is a subspace of \(V\) if and only if either \(U \subseteq W\) or \(W \subseteq U\).