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Homework 3
- Solution Manual: Chapter 3
- Section 3.2
- Watch Lecture 7: Solving Ax=0: Pivot Variables, Special Solutions of the MIT OpenCourseWare series
- In Class: 7*:
- 7*. Follow the directions for Problem 7 in Problem Set 3.2, with (a) 2, 3, 6, 7 and (b) 1, 2, 4, 5, 7
- Homework: 1-9, 11, 13, 14, 17, 18, 20-22, 24, 32, 33, 39, 41
- Hand In: 7**, 13*, 17*:
- 7**. Follow the directions for Problem 7 in Problem Set 3.2, with (a) 2, 3, 4, 8 and (b) 1, 3, 4, 5, 8
- 13*. Follow the directions for Problem 13 in Problem Set 3.2, with the planes \(2x+y-4z=8\) and \(2x+y-4z=0\), and the appropriate adjustment to one particular point on the second plane.
- 17*. Follow the directions for Problem 17 in Problem Set 3.2, with the column space containing \((1,-1,2)\) and \((2,2,0)\) and the null space containing \((1,1,-1)\). (Find a \(3 \times 3\) matrix.)
- Section 3.3
- Watch Lecture 8: Solving \(A{\bf x}={\bf b}\): row reduced form \(R\) of the MIT OpenCourseWare series
- In class: 1*:
- 1*. Follow the directions for Problem 1 in Problem Set 3.3, with \[A = \left[\matrix{1&-1&4&2\cr 2&-1&7&-1\cr 3&-1&10&-4}\right]\] and set \[{\bf b} = \left[\matrix{b_1\cr b_2\cr b_3}\right]\ \mbox{equal to}\ \left[\matrix{3\cr -1\cr -5}\right]\ \mbox{in Parts 5 and 6.}\]
- Homework: 2, 1, 4, 6, 7, 8, 11, 13, 14, 16, 18, 20, 22, 30, 29
- Hand In: 1**, 20*, 29*:
- 1**. Follow the directions for Problem 1 in Problem Set 3.3, with \[A = \left[\matrix{2&3&-1&2\cr 4&8&-4&6\cr 6&17&-11&14}\right]\] and set \[{\bf b} = \left[\matrix{b_1\cr b_2\cr b_3}\right]\ \mbox{equal to}\ \left[\matrix{-1\cr -2\cr -3}\right]\ \mbox{in Parts 5 and 6.}\]
- 20*. Follow the directions for Problem 20 in Problem Set 3.3, with \[A = \left[\matrix{2&-1&2&3\cr -4&4&-3&0}\right]\ \mbox{and}\ A = \left[\matrix{1&0&1&1\cr -3&2&-2&-4\cr 0&4&7&6}\right].\]
- 29*. Follow the directions for Problem 29 in Problem Set 3.3, with \[U = \left[\matrix{2&0&8&0\cr 0&0&4&0\cr 0&0&0&0}\right]\ \mbox{and}\ {\bf c} = \left[\matrix{5\cr 2\cr 0}\right].\] (The system (three equations, four unknowns) is first augmented with a zero right hand side, and then with \({\bf c}\) as right hand side. The number of free variables will be \(2\), not \(1\).)
- Section 3.4
- Watch Lecture 9: Independence, Basis and Dimension of the MIT OpenCourseWare series
- In class: 16*:
- 16*. Find a basis for each of these subspaces of \(\mathbb{R}^4\):
- (a) All vectors whose first three components are equal.
- (b) All vectors whose first two components add to zero.
- (c) All vectors that are perpendicular to \((1,0,0,1)\ \mbox{and}\ (1,1,1,0)\).
- (d) The column space and the nullspace of the \(4 \times 4\) identity matrix \(I\).
- 16*. Find a basis for each of these subspaces of \(\mathbb{R}^4\):
- Homework: 2, 1, 3-5, 7-12, 15, 16, 18, 19, 24, 25, 26, 31, 37, 38, 41
- Hand In: 7*, 16**, 38*:
- 7*. Follow the directions for Problem 7 in Problem Set 3.4, with \({\bf v}_1 = {\bf w}_1+{\bf w}_2+{\bf w}_3, {\bf v}_2 = {\bf w}_1-{\bf w}_2, {\bf v}_3 = -2{\bf w}_2-{\bf w}_3\).
- 16**. Find a basis for each of these subspaces of \(\mathbb{R}^4\):
- (a) All vectors whose first two components are equal.
- (b) All vectors whose first three components add to zero.
- (c) All vectors that are perpendicular to \((1,0,1,0)\ \mbox{and}\ (1,1,0,1)\).
- 38*. Which of the following are bases of \(\mathbb{R}^4\)? (Give reasons!)
- \((1,2,3,4), (5,6,7,8), (9,0,1,2), (3,4,5,6), (7,8,9,0)\)
- \((1,2,3,4), (5,6,7,8), (9,0,1,2)\)
- \((1,2,3,4), (5,6,7,8), (9,0,1,2), (3,4,5,6)\)
- \((1,2,3,4), (5,6,7,8), (9,0,1,2), (3,-8,-9,-10)\)
- Section 3.5
- Watch Lecture 10: The Four Fundamantal Subspaces and Lecture 11: Matrix spaces; rank 1; small world graphs of the MIT OpenCourseWare series
- In class: 3*:
- 3*. Follow the directions for Problem 3 in Problem Set 3.5, with \[A = \left[\matrix{1&2&3&4&5\cr -1&-2&-2&-2&-2\cr 0&0&-1&-2&-3}\right] = \left[\matrix{1&0&0\cr -1&1&0\cr 0&-1&1}\right]\left[\matrix{1&2&3&4&5\cr 0&0&1&2&3\cr 0&0&0&0&0}\right]\]
- Homework: 1, 3, 4, 6, 7, 9, 11, 13, 16, 17, 18, 20, 21, 23-27
- Hand In: 3**, 17*, 25*:
- 3**. Follow the directions for Problem 3 in Problem Set 3.5, with \[A = \left[\matrix{1&2&3&4&5\cr 1&2&4&6&8\cr 0&0&1&3&5\cr 0&0&0&1&2}\right] = \left[\matrix{1&0&0&0\cr 1&1&0&0\cr 0&1&1&0\cr 0&0&1&1}\right]\left[\matrix{1&2&3&4&5\cr 0&0&1&2&3\cr 0&0&0&1&2\cr 0&0&0&0&0}\right]\]
- 17*. Follow the directions for Problem 17 in Problem Set 3.5, with \[A = \left[\matrix{0&0&1\cr 0&0&0\cr 0&0&0}\right]\] (Adjust the matrix \(I+A\) appropriately)
- 25*. Follow the directions for Problem 25 in Problem Set 3.5, with
- (a) \(A\) and \(A^t\) have the same number of non-pivots.
- (b) \(A\) and \(A^t\) have the same nullspace.
- (c) \(A\) may have the same row and column spaces.