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Homework 6
- Section 3.2 Matrix Multiplication (Composition)
- Read Section 3.2. Do the True/False questions (p. 171). Do Problems 3.25, 3.26.
- Homework: 3.28, 3.29, 3.30, 3.34, 3.33 (the unit square has vertices \((0,0), (1,0), (1,1)\) and \((0,1)\)), 3.41, 3.45, 3.48, 3.50, 3.53 (do Part (c) in general, that is, for *any* vector P in \(\mathbb{R^n}\)), 3.55, 3.57
- Hand In: 3.33* (the unit square has vertices \((0,0), (1,0), (1,1)\) and \((0,1)\)), 3.41*, 3.53* (do Part (c) in general, that is, for *any* vector \(P\) in \(\mathbb{R^n}\))
- 3.33*. Complete Problem 3.33 on Page 172, but where \(T(X)\) is the result of first rotating \(X\) counterclockwise by \(2\pi/3\) and then multiplied by \[A=\left[\matrix{4&0\cr 0&3\cr}\right].\]
- 3.41*. Complete Problem 3.41 on Page 174, but where \(C\) is the matrix \[C=\left[\matrix{2&2&-1\cr -1&1&2\cr 1&3&1\cr}\right].\]
- 3.53*. Complete Problem 3.53 on Page 175, but with \(P=[1,2,-2]^t\).
- Read Section 3.3. Do the True/False questions (pp. 191-192). Do Problem 3.68.
- 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 \[\left[\matrix{1&0&1&0\cr 0&1&0&0\cr -1&0&1&1\cr 1&1&1&1\cr}\right]\]
- 3.79*. Complete Problem 3.79 on Page 194, but where \[B=\left[\matrix{1&2&2\cr 1&-2&1\cr 0&1&3\cr}\right]\ \mbox{and}\ C=\left[\matrix{2&1&3\cr -1&0&1\cr 2&1&-1\cr}\right] \]
- 3.82*. Complete Problem 3.82 on Page 195, but with the matrix \(A\) satisfying the equation \(A^3-2A^2+5A-I={\bf 0}\).
- 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)).