Linear Algebra : Reading + Assignment 1

Reading

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.

 

Assignment 1

Problem 1.  Write down the (augmented) matrix that represents the system of equations

\begin{array}{cccccc}2x_1 &+& 3x_2 &+& x_3 &= 1 \\x_1 & - & x_2 & + & x_3 &= 0\end{array}

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

\left(\begin{array}{cccc|c}1 & 1 & 1 & 1 & 0 \\ -1 & 2 & -2 & 0 & 0 \\ 0 & 3 & -1 & 1 & 0\end{array}\right).

Problem 4.  Solve the system of equations that corresponds to the augmented matrix

\left(\begin{array}{cccc|c}1 & 1 & 1 & 1 & 1 \\ -1 & 2 & -2 & 0 & 1 \\ 0 & 3 & -1 & 1 & 2\end{array}\right).

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)

 

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