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Homework 4
- Section 3.1
- Read Section 3.1.
- Watch Lecture 6: Column Space and Null Space of the MIT OpenCourseWare series
- Worked examples (group work): 5, 17*.
- 5. As in the textbook.
- 17*. Follow the directions for Problem 17 in Problem Set 3.1, with the matrices \[ A=\left[\matrix{1&-3&2\cr 3&-9&6\cr -2&6&-4}\right]\ \mbox{and}\ A=\left[\matrix{1&3\cr 2&7\cr -1&-3}\right].\]
- Homework: 1, 3, 9-13 odd, 17, 19-22, 25, 31
- Hand In: 17**, 25*, 31:
- 17**. Follow the directions for Problem 17 in Problem Set 3.1, with the matrices \[ A=\left[\matrix{1&-2&4\cr 3&-6&12\cr -2&4&-8}\right]\ \mbox{and}\ A=\left[\matrix{1&-2\cr 3&-5\cr 2&-4}\right].\]
- 25*. Follow the directions for Problem 25 in Problem Set 3.1, with the vectors \( (1,2,0)\ \mbox{and}\ (1,0,2)\) in the subspace and the vector \((0,1,1)\) not in the subspace.
- 31. As in the textbook.
- Section 3.2
- Read Section 3.2.
- Watch Lecture 7: Solving Ax=0: Pivot Variables, Special Solutions of the MIT OpenCourseWare series
- Worked example (group work): 25*.
- 25*. What are all the \(2 \times 3\) reduced matrices \(R\) whose entries are all \(0\)s and \(1\)s? Choose one of these matrices and find its null space. (In your copious free time, find the null spaces of all of these matrices.)
- Homework: 1, 2, 4, 5, 7, 9, 13, 15, 19, 25, 31-35 odd, 39, 45
- Hand In: 4, 13, 35
- In Problem 4 Strang means reduce \(A\) and \(B\) to their reduced row echelon forms \(R_0\), then determine the \(CR\) factorizations.
- In Problem 13 you may choose to use parameters (e.g., \(s\mbox{ and } t\)) instead of retaining \(y\mbox{ and }z\) in your solution.
- In Problem 35 there is an error in the matrix \(A^T\). From the circuit diagram at the left, you can see that the current expression at Node 3 should be \(y_2-y_3-y_6 = 0\). Correct the third row of the matrix \(A^T\) accordingly (change the sign of the final entry of Row 3). Your reduction will first be to \(R_0\), which has four rows.