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Homework 3
- Section 1.4 Column Space and Null Space
- Read Section 1.4. Do the True/False questions (pp. 81-82). Do Problem 1.180.
- 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 \(\mathcal{W}=\ \mbox{span}\{[1,0,1]^t,[0,1,1]^t\}\)
- 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 \[[2s+2t-2u,-2s-4u,3s+t+3u]^t.\]
- 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 \(f^{\prime\prime}(-1)=0\).
- 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 \(\mathcal{V}\).