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Homework 2
- Section 1.4 Column Space and Null Space
- Read Section 1.4. Do the True/False questions (pp. 81-82). Do Problem 1.180. WeBWorK reading question due February 1 @ 9:50 am.
- In class (group work): 1.83, 1.85(a) (Read "Create an m x n matrix" (not "Create an n x m matrix")), 1.88. Final due date: February 15
- Homework: 1.84, 1.85(c) (Read "Create an m x n matrix" (not "Create an n x m matrix")), 1.87, 1.89, 1.91, 1.94, 1.93, 1.96(b)(c)(d), 1.101, 1.102, 1.106, 1.108, 1.109, 1.113, 1.114, 1.115
- Hand In: 1.85(c)* (Read "Create an m x n matrix" (not "Create an n x m matrix")), 1.96(d)*, 1.109*, 1.113*
- 1.85(c)*. Complete Problem 1.85(c) on Page 83, but with \(m\times n = 3 \times 4\) and \(\mathcal{W}=\ \mbox{span}\{[1,1,0]^t,[0,1,1]^t\}\)
- 1.96(d)*. Complete Problem 1.96(d) on Pages 85-87, but where \(\mathcal{W}\) is the set of points in \(\mathbb{R}^3\) of the form \[[2s+2t-u,-2s-2t+u,6s+2t]^t.\]
- 1.109*. Complete Problem 1.109 on Page 89, but where \(\mathcal{W}\) is the set of all functions \(f\in C^\infty(\mathbb{R})\) such that \(f^{\prime\prime}(2)=0\).
- 1.115*. Similar to Problem 1.115 on Page 89: Let \(\mathcal{S}\) and \(\mathcal{T}\) be subspaces of a vector space \(\mathcal{V}\). Let \(\mathcal{U} = \{X+Y\ \vert\ X\in \mathcal{S}, Y\in \mathcal{T}\}\). Prove that \(\mathcal{U}\) is a subspace of \(\mathcal{V}\).
- Read Section 2.1. Do the True/False questions (p. 104). Do Problems 2.1(a)(d)(g)(j). WeBWorK reading question due February 6 @ 9:50 am.
- In class (group work): 2.1(c), 2.3(b) (there's a typo: use the technique of Example 2.4). Final due date: February 20
- Homework: 2.1(b)(e)(f)(k), 2.3(a)(c)(d)(f), 2.5, 2.6, 2.9(a)(c), 2.11, 2.13, 2.16(a)(c)(e), 2.17(a)(c)(d)(f), 2.19
- Hand In: 2.1(k)*, 2.3(f)*, 2.9(c)*, 2.17(d)*
- 2.1(k)*. Complete Problem 2.1(k) on Page 105, but with the vectors \[\left[\matrix{3\cr 2\cr 4\cr 5\cr}\right], \left[\matrix{-4\cr -3\cr -5\cr -6\cr}\right], \left[\matrix{6\cr 3\cr 9\cr 12\cr}\right]\]
- 2.3(f)*. Complete Problem 2.3(f) on Pages 105-106, but with the matrix \[\left[\matrix{3&6&2&-7\cr 2&4&0&8\cr -1&-2&-1&-2\cr 2&4&-3&8\cr}\right]\]
- 2.9(c)*. Complete Problem 2.9(c) on Page 107, but with \(Y_1=X_1+3X_2-X_3, Y_2=X_1-X_2\) and \(Y_3=5X_2\).
- 2.17(d)*. (Edited due to online version of text showing a random matrix instead of sets of functions:) Show that the set \(\{x^3,x^4,x^5\}\) (of infinitely differentiable functions) is linearly independent using the test for linear independence.
- Read Section 2.2. Do the True/False questions (p. 118). Do Problems 2.20, 2.22, 2.24(a). WeBWorK reading question due February 13 @ 9:50 am.
- In class (group work): 2.24(a), 2.27(a), 2.30(a). Final due date: February 27
- Homework: 2.24(b)(c), 2.25, 2.26, 2.27, 2.29, 2.30, 2.33, 2.35, 2.36, 2.38, 2.41, 2.43, 2.45
- Hand In: 2.24(c)*, 2.27(c)*, 2.30(c)*, 2.38*
- 2.24(c)*. Complete Problem 2.24(c) on Page 119, but with \[\mathcal{W}=\{[a+b+2c,2a+b+3c,a+2b+c]^t\ \vert\ a, b, c \in \mathbb{R}\}.\]
- 2.27(c)*. Complete Problem 2.27(c) on Page 119, but where A is lower triangular (and \(3\times 3\)).
- 2.30(c)*. Complete Problem 2.30(c) on Page 120, but with the matrix \[\left[\matrix{1&-5&2&3\cr 2&-10&4&6\cr 4&-20&8&12\cr}\right]\]
- 2.38*. Complete Problem 2.38 on Page 121, but with \(X = 2A-3B\) and \(Y = 3A+5B.\)