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Homework 3
- Solution Manual: Chapter 2; Chapter 3
- Section 2.7
- Read Section 2.7. WeBWorK reading question due February 6 @ 8:50 am.
- Watch Lecture 5: Transposes, permutations, spaces \({\mathbb{R}}^n\) of the MIT OpenCourseWare series
- In class: 11*. Final due date: February 22.
- 11*. Follow the directions for Problem 11 in Problem Set 2.7, with the matrix \[A = \left[\matrix{0&-2&1\cr 0&0&3\cr 1&-1&2}\right]\]
- Homework: 1, 2, 4, 7, 11, 15, 17, 18, 20, 21, 24, 26, 30, 32
- Hand In: 11**, 21*, 26*:
- 11**. Follow the directions for Problem 11 in Problem Set 2.7, with the matrix \[A = \left[\matrix{0&0&-2&1\cr 1&-1&2&3\cr 0&0&0&-2\cr 0&3&4&2}\right]\]
- 21*. Follow the directions for Problem 21 in Problem Set 2.7, with the matrices \[S = \left[\matrix{1&3&-5\cr 3&2&2\cr -5&2&7}\right]\ \mbox{and}\ S = \left[\matrix{-1&b&c\cr b&d&e\cr c&e&f}\right]\]
- 26*. Follow the directions for Problem 26 in Problem Set 2.7, with the matrices \[S = \left[\matrix{1&-2&0\cr -2&7&6\cr 0&6&10}\right]\ \mbox{going toward}\ D = \left[\matrix{1&0&0\cr 0&3&0\cr 0&0&-2}\right].\] (You'll need to use different multipliers, that is, in (a), remove the \(-2\) below the pivot; in (b), remove the \(6\) below the pivot)
- Section 3.1
- Read Section 3.1. WeBWorK reading question due February 8 @ 8:50 am.
- Watch Lecture 6: Column Space and Null Space of the MIT OpenCourseWare series
- In Class: 10*. Final due date: February 24.
- 10*. Which of the following subsets of \(\mathbb{R}^3\) are actually subspaces of \(\mathbb{R}^3\)?
- (a) The plane of vectors \((b_1,b_2,b_3)\) with \(b_3 = b_1+b_2\).
- (b) The plane of vectors \((b_1,b_2,b_3)\) with \(b_2 = -1\).
- (c) The vectors \((b_1,b_2,b_3)\) with \(b_1b_2=0\).
- (d) All linear combinations of \({\bf v} = (-1,2,3)\) and \({\bf w} = (0,1,4)\).
- (e) All vectors \((b_1,b_2,b_3)\) satisfying \(b_1+b_2+b_3=3\).
- (f) All vectors \((b_1,b_2,b_3)\) with \(b_1 \leq b_3\).
- 10*. Which of the following subsets of \(\mathbb{R}^3\) are actually subspaces of \(\mathbb{R}^3\)?
- Homework: 1, 3, 7, 10, 11, 15, 16, 17, 19, 20, 22, 23, 25, 28
- Hand In: 10**, 22*, 28*:
- 10**. Which of the following subsets of \(\mathbb{R}^3\) are actually subspaces of \(\mathbb{R}^3\)?
- (a) The plane of vectors \((b_1,b_2,b_3)\) with \(b_3 = b_1+1\).
- (b) The plane of vectors \((b_1,b_2,b_3)\) with \(b_3 = 0\).
- (c) The vectors \((b_1,b_2,b_3)\) with \(b_2b_3=1\).
- (d) All linear combinations of \({\bf v} = (-1,-1,2)\) and \({\bf w} = (-2,2,-4)\).
- (e) All vectors \((b_1,b_2,b_3)\) satisfying \(b_1+2b_2+3b_3=0\).
- (f) All vectors \((b_1,b_2,b_3)\) with \(b_1 \geq b_3\).
- 22*. Follow the directions for Problem 22 in Problem Set 3.1, with the matrices \[ \left[\matrix{1&0&1\cr 0&1&0\cr 0&0&1}\right], \left[\matrix{1&0&1\cr 0&1&0\cr 0&0&0}\right]\ \mbox{and}\ \left[\matrix{1&1&1\cr 0&1&1\cr 0&1&1}\right]\]
- 28*. Follow the directions for Problem 28 in Problem Set 3.1, with the column space instead containing \((-1,2,-3)\) and \((1,1,-1)\) but not \((0,3,1)\). For the second part make the column space be the line containing the vector \((1,1,-1)\).
- 10**. Which of the following subsets of \(\mathbb{R}^3\) are actually subspaces of \(\mathbb{R}^3\)?
- Section 3.2
- Read Section 3.2. WeBWorK reading question due February 17 @ 8:50 am.
- Watch Lecture 7: Solving Ax=0: Pivot Variables, Special Solutions of the MIT OpenCourseWare series
- In Class: 7*. Final due date: March 1.
- 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 first 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
- Read Section 3.3. WeBWorK reading question due February 20 @ 8:50 am.
- Watch Lecture 8: Solving \(A{\bf x}={\bf b}\): row reduced form \(R\) of the MIT OpenCourseWare series
- In class: 1*. Final due date: March 3.
- 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
- Read Section 3.4. WeBWorK reading question due February 24 @ 8:50 am.
- Watch Lecture 9: Independence, Basis and Dimension of the MIT OpenCourseWare series
- In class: 16*. Final due date: March 6.
- 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!)
- \(\mathcal{B}_1=\{(1,2,3,4), (5,6,7,8), (9,0,1,2), (3,4,5,6), (7,8,9,0)\}\)
- \(\mathcal{B}_2=\{(1,2,3,4), (5,6,7,8), (9,0,1,2)\}\)
- \(\mathcal{B}_3=\{(1,2,3,4), (5,6,7,8), (9,0,1,2), (3,4,5,6)\}\)
- \(\mathcal{B}_4=\{(1,2,3,4), (5,6,7,8), (9,0,1,2), (3,-8,-9,-10)\}\)