For this assignment you should read Chapter 1, Sections I, II, and III. There is a lot of material in there, and I think it would be most helpful to follow all of the examples. This reading (and your notes from class) should help you complete your first assignment.
There are a few chucks of these sections that you should feel free to skim through or skip all together. In particular, I recommend doing this starting at the bottom of page 26 (at the word “proof”), and ending towards the end of page 27 (at the phrase “QED”). I also recommend skipping the proof of Lemma 3.7 and, instead, focusing on relating that result to Examples 3.8 and 3.9.
Problem 1. Write down the (augmented) matrix that represents the system of equations
Is this a homogeneous system? Why or why not?
Problem 2. What does it mean to say that a system is inconsistent? Explain whether or not it is possible for a homogeneous system to be inconsistent.
Problem 3. Solve the system of equations that corresponds to the augmented matrix
Problem 4. Solve the system of equations that corresponds to the augmented matrix
Problem 5. Compare your answers to Problems 3 and 4 to the prediction made by Lemma 3.7 (pg. 27). Can your solution set for Problem 4 be written according to this result? If so, do it! If not, um, spoiler alert, but you’ve made a mistake!
Problem 6. What is a leading variable in a system of equations? What is a free variable?
Problem 7. Complete Problem 2.18(e) (pg. 20).
Problem 8. Complete Problem 2.22(a) (pg. 21)
Problem 9. Complete Problem 3.16(a) (pg. 33)
Problem 10. 3.19 (pg. 33)