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Homework 5
- Section 2.3 Row Space and the Rank-Nullity Theorem
- Read Section 2.3. Do the True/False questions (pp. 139-140). Do Problem 2.56. WeBWorK reading question due February 16 @ 11:50 am.
- Homework: 2.57(d) (for the matrix you'll have to do something different than the problem suggests to find the different bases the problem asks for), 2.58, 2.61, 2.64, 2.68, 2.69, 2.71, 2.72, 2.73
- Hand In: 2.57(d)*, 2.69*, 2.71*, 2.73*
- 2.57(d)*. Complete Problem 2.57(d) on Page 140, but with the matrix \[\left[\matrix{1&2&1&2\cr 2&4&2&4\cr -1&-2&-1&2\cr -3&-6&-3&6\cr}\right]\]
- 2.69*. Replace Problem 2.69 in the text with the following. Let \(A\) be a non-singular matrix. What is the dimension of the nullspace of \(A\)? What is the dimension of the nullspace of \(A^t\)?
- 2.71*. Replace Problem 2.71 in the text with the following. Let \(A\) be an \(m\times n\) matrix of rank \(m\). Show that \(m\leq n\). What is the dimension of the nullspace of \(A\)?
- 2.73*. Replace Problem 2.73 in the text with the following. Let \(A\) be an \(m\times n\) matrix of rank \(n\). Show that \(m\geq n\). How many zero rows does a row echelon form for \(A\) have?
- WeBWorK assignment for Section 2.3 due February 26 @ 11 pm
- Read Section 3.1. Do the True/False questions (p. 154). Do Problems 3.1, 3.3. WeBWorK reading question due February 23 @ 11:50 am.
- Homework: 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.13, 3.14, 3.16, 3.18, 3.19, 3.20, 3.21, 3.22, 3.23 (typo in last sentence: "Show that the image of \(T\) is a subspace of \(\mathcal{W}\)" (not of \(\mathcal{V}\))
- Hand In: 3.6*, 3.14(e)*, 3.18*, 3.23* (typo in last sentence: "Show that the image of \(T\) is a subspace of \(\mathcal{W}\)" (not of \(\mathcal{V}\))
- 3.6*. Complete Problem 3.6 on Page 156, but with \(A\) transforming the parallelogram with vertices \((0,0), (2,2),(3,5),(1,3)\) onto the parallelogram with vertices \((0,0),(1,0),(2,1),(1,1)\) (the right figure of Figure 3.5).
- 3.14(e)*. Complete Problem 3.14(e) on Page 158, but with the transformation \[T([x,y,z]^t)=[4x-2y+z,2y-3z]^t\]
- 3.18*. Complete Problem 3.18 on Page 159, but with \(T, S, U : C([-1,2])\to\mathbb{R}\) as follows. \[\mbox{(a)}\ T(f)=\int_{-1}^2f(x)\ dx\] \[\mbox{(c)}\ S(f)=\int_{-1}^2(f(x))^2\ dx\] \[\mbox{(e)}\ U(f)=\int_{-1}^2xf(x)\ dx\]
- 3.23*. Complete Problem 3.23 on Page 160, but altered to read as follows. Let \(T\ :\ \mathcal{V}\to\mathcal{W}\) be a linear transformation between two vector spaces. Suppose that \(\mathcal{U}\) is a subspace of \(\mathcal{V}\). Define the image of \(\mathcal{U}\), denoted \(T(\mathcal{U})\) to be \(T(\mathcal{U})= \{T(X)\ \vert\ X \in \mathcal{U}\}\). Show that \(T(\mathcal{U})\) is a subspace of \(\mathcal{W}\).