MAT 2240 — §101. Spring '17 (171).

Let us reflect that, having banished from our land that religious intolerance under which mankind so long bled and suffered, we have yet gained little if we countenance a political intolerance as despotic, as wicked, and capable of as bitter and bloody persecutions.

— Thomas Jefferson (1743-1826) in his First Inaugural Address.

Homework List

✈ Jump down to '♞ This Week.'

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Week 1

• Monday, Jan 16
  ◊ Martin Luther King, Jr., Day — no classes
  ¤ Rarely do we find men who willingly engage in hard, solid thinking. There is an almost universal quest for easy answers and half-baked solutions. Nothing pains some people more than having to think. — Dr. Martin Luther King, Jr. (1929-1968)
  
  
• Wednesday, Jan 18
  ◊ Read §1.1, 1.2.
  §1.1. Pg 10; #1, 2, 6, 11, 15, 17, 29, 31, 33, 34.
  ¤ Check out Prof Bogacki's online “Linear Algebra Toolkit”
  
  
• Friday, Jan 20
  §1.2. Pg 21; #1, 2.
  
  

Week 2

• Monday, Jan 23
  §1.2. Pg 21; 3, 9, 13, 19, 21, 23, 29, 33, (for CS or Computational Math majors:) 32.
  
  
• Wednesday, Jan 25
  §1.3. Pg 32; #5, 9, 11, 13, 15, 19, 21, 26, 27, LA 28 ("LA" is "Look at"), (for Physics or Applied Math majors:) 29.
  
  
• Friday, Jan 27
  ¡Quiz next Friday!
  §1.4. Pg 40; #1, 2, 4, 5, 7, 11, 13, 15, 17, 21.
  
  

Week 3

• Monday, Jan 30
  ❄ All classes before 10 am Noon cancelled.
  ¤ Thm 4, pg 37, is very important! Study it carfully.
  §1.4. Pg 40; #3, 8, 9, 19, 23, 25, 29, 39.
  ☝ Good quiz questions come from #23.
  ◊ For #39: Start Maple. Click on "New Document."
  1. Load the Student Linear Algebra package using the Load Package submenu of the Tools menu.
  2. Define the matrix \(A\) using the Matrix palette.
  3. You can now use the command ReducedRowEchelonForm(A).
  
  
• Wednesday, Feb 1
  §1.5. Pg 47; #1, 3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 28 → 31.
  
  
• Friday, Feb 3
  ¡Quiz today!
  §1.6. Pg 54; #1, 7, 12.
  ¤ Hints for #12:
  1. One equation for each node: \(A\), \(B\), and \(C\).
  2. Flow into a node must equal flow out.
  3. All flows must be \(\ge 0\) gives a restriction on \(x_4\).
  
  

Week 4

• Monday, Feb 6
  §1.6. Pg 54; #10, 11, 13.
  
  
• Wednesday, Feb 8
  §1.7. Pg 60; #1, 3, 5, 9, 11, 15, 17, 21, 27, 42.
  
  
• Friday, Feb 10
  ¡It's time to think about our first midterm exam — next Friday!
  ◊ The problems on the Linear Independence Worksheet
  
  

Week 5

• Monday, Feb 13
  §1.8. Pg 68; #1, 2, 3, 5, 7, 8, 9, 11, 15, 16, 20, 21, 39.
  
  
• Wednesday, Feb 15
  ◊ For Monday: §1.9. Pg 78; #1, 3, 5, 11, 13, 15, 16, 17, 21, 25, 27, 40.
  
  
• Friday, Feb 17
  ◊ Test I is today — A chance to demonstrate your excellence!
  • You can bring one 8½"×11" sheet of notes.
  • You can bring a calculator; check the batteries!
  ¤ Test I Topics List
  
  Saturday is Elm Farm Ollie Day
  
  

Week 6

• Monday, Feb 20
  §2.1. Pg 100; #1, 3, 7, 8, 9, 11, 17, 21, 40.
  ◊ Maple project: Resistor Networks. Due Friday.
  
  
• Wednesday, Feb 22
  §2.2. Pg 109; #1, 3, 5, 8, 9, 13, 21, 31, 33.
  ◊ Extend Pg 100, #40: Use the Maple statements (copy below, then paste into Maple )
  • gen := (i,j) -> piecewise(i+1=j, 1, 0):
    T := n -> Matrix(n,n, gen):
    S := T(5);
     to make a function \(T\) that produces square matrices with the given pattern of 1's.
     #40e. Replace the 5 with several larger values to discover a pattern in the powers \(S^k\) for \(k=1, 2, \dots\).
     #40f. Does \(S(n)\) have an inverse?
  
  
• Friday, Feb 24
  §2.3. Pg 115; #1 → 9 odd, 11, 13, 14, 18, 19, 33, 36.
  
  

Week 7

• Monday, Feb 27
  §2.5. Pg 129; $1, 3, 5, 7, 13, 17.
  ¤ Maple has a command LUDecomposition.    Remember to load Student[LinearAlgebra] first.
      Enter: ? command to get help with any command.
  
  
• Wednesday, Mar 1
  ¤ Read either §2.6 (economics) or §2.7 (computer graphics) as your interestx dictate.
  §2.8. Pg 151; #1 → 4, 5, 7, 9.
  
  
• Friday, Mar 3
  ¡Quiz today! — A chance to demonstrate your excellence!
  §2.8. Pg 151; #11, 13, 17, 19, 21, 25, 27, 29, 37.
  
  

Week 8

• Monday, Mar 6
  §2.8. Pg 151; #23, 24, 28, 31, 33, 35, 38.
  
  
• Wednesday, Mar 8
  §2.9. Pg 157; #1, 3, 5, 7, 11, 13, 15.
  
  
• Friday, Mar 10
  §2.9. Pg 157; #17, 18, 19, 21, 25, 27, 29.
  
  

Spring Break Week — Mar 13 → Mar 17: «Hey, let's be careful out there!»


Week 9

• Monday, Mar 20
  ¤ Chapter 2 True/False
  § Chapter 2 Supplementary Exercises. Pg 160; the odd problems.
  
  
• Wednesday, Mar 22
  §3.1. Pg 167; #1, 3, 5, 13, 19, 21, 23, 24, 25, 27, 29, 33, 39, 43, 45, 46.
  ◊ For #45 & 46. Maple: A:=RandomMatrix(4,4, generator=-9..9). Two useful functions are Determinant and Transpose
  
  
• Friday, Mar 24
  §3.2. Pg 175; #1, 3, 7, 10, 15, 17, 19, 23, 25, 46 (cf. §2.3, Ex 9, pg. 115).
  ◊ For #45. In Maple, define a function for the condition number of a matrix with: CN := LinearAlgebra[ConditionNumber]:
     Then CN(A) will give the condition number of the matrix \(A\).
  ¤ Predict the 11th row in the data for finding a determinant using the definition: \[ \begin{array}{r|rr} n & ArithOps & Time \\ \hline 1 & 0 & 0.001 \\ 2 & 3 & 0.001 \\ 3 & 17 & 0.001 \\ 4 & 95 & 0.002 \\ 5 & 599 & 0.007 \\ 6 & 4319 & 0.086 \\ 7 & 35279 & 0.493 \\ 8 & 322559 & 4.961 \\ 9 & 3265919 & 68.322 \\ 10 & 36287999 & 1616.794 \\ 11 & ?\quad & ?\quad \end{array} \]
  

Week 10

• Monday, Mar 27 — is the last day to drop a class.
  §4.1. Pg 195; #1, 5 → 8, 9, 13, 19, 21, 24, 27. Look at: 32, 33.
  
  
• Wednesday, Mar 29
  §4.2. Pg 205; #1, 5, 7, 13, 25, 26.
  ¤ Vector space slides
  
  
• Friday, Mar 31 — Experimental class: all online.
  ¤ Read:
  • Markov chain
  • Markov Chains
  • Markov Chains
  ¤ Watch:
  • Transition Matrices
  • Transition & Initial State Matrices
  §4.9. Pg 260; #1, 4, 5, 7.
  ¤ Analyze the state diagram for ASU using an initial class of \(2,\!500\) freshman.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  1. How long before all the freshman have either graduated or dropped out?
  2. What is the `four-year graduation rate'? `Five-year graduation rate'? `Six-year graduation rate'?
  3. What are the steady state class sizes if \(2,\!500\) freshman are admitted each year?
  4. How could transfer students be added to the model?

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  ¡It's time to think about our next midterm exam — next Friday!
  
  

Week 11

• Monday, Apr 3
  ◊ Project Day (no formal class meeting
  ¤ Maple-based group project (2 or 3 people): Matrix Algebra And Modular Arithmetic: Hill Codes (LIN06) — Due: 4/10/17.
  
  
• Wednesday, Apr 5 — Experimental class: all online.
  ◊ Review:
      § Chapter 2 Supplementary Exercises.
      § Chapter 3 Supplementary Exercises.
      § Chapter 4 Supplementary Exercises.
  ◊ Khan Academy Linear Algebra Videos
  
  
• Friday, Apr 7
  ◊ Test II is today — A chance to demonstrate your excellence!
  ¤ Since I have not been able to access the website or email while I was away from Boone due to some unexplained tech-glitch,
       the test is postponed until Monday, April 10.
  
  

Week 12

• Monday, Apr 10
  ◊ Test II is today — A chance to demonstrate your excellence!
  • You can bring two 8½"×11" sheets of notes.
  • You can bring a calculator; check the batteries!
  ¤ Test II Topics List
  
  
• Wednesday, Apr 12
  §5.1. Pg 271; #1, 3, 5, 9, 15, 19, 21, 29.
  
  
• Friday, Apr 14
  §5.2. Pg 279; #1→9 odd, 17, 18, 28.
  
  

Week 13

• Monday, Apr 17
  ¡No classes Monday or Tuesday! — Ēostre Holiday
  
  
• Wednesday, Apr 19
  §5.2. Pg 279; #19, 21, 23, 33, 39.
  
  
• Friday, Apr 21
  ¡Quiz next Friday!
  ◊ Eigenspaces slides
  ¤ Problems:
  1. Determine the algebraic and geometric dimensions of each eigenvalue in the matrices: \[ A = \begin{bmatrix} 0 & -12 & -3 & 3 \\ -6 & 6 & 1 & -5 \\ 0 & 0 & -2 & 4 \\ 0 & 12 & 8 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & -6 & 1 & 1 \\ 0 & 3 & -1 & 0 \\ -1 & -1 & 0 & 0 \\ -2 & 10 & -1 & -1 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & -2 & 1 & -2\\ 2 & 2 & 0 & 3 \\ 2 & 4 & -1 & 5 \\ 0 & 3 & -2 & 2 \end{bmatrix} \]
  2. Using Maple, find both the algebraic and geometric dimensions of all the eigenvalues for 5 random matrices generated by:
          A := RandomMatrix(6,6, generator=-1..2, shape=triangular);
          cp := CharacteristicPolynomial(A,x);
          ev := fsolve(cp=0, x, complex);
          evalf(Eigenvectors(A, output=list), 2);
          (You can copy and paste the Maple statements into a Maple worksheet.)
       • Make a conjecture about how often the geometric dimension is smaller than the algebraic dimension for this type of matrix.
  
  

Week 14

• Monday, Apr 24
  ¤ Use the Müntz power method to find the dominant eigenvector for the matrix \[ \begin{bmatrix} 1 & 4 & 9 & 3 \\ 1/4 & 1 & 3 & 1/5 \\ 1/9 & 1/3 & 1 & 1/2 \\ 1/3 & 5 & 2 & 1 \end{bmatrix}, \] and scale the eigenvector to become a percentage vector for an AHP problem.
  ◊ Two potential quiz questions:
  
       Given \(B = \begin{bmatrix} -1 & -1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & -2 \end{bmatrix} \) and \(\bar{x} = \begin{bmatrix} -1 \\ 3 \\ 1 \end{bmatrix}\).
  
  
  1. Is \(\bar{x}\) an eigenvector of \(B\)?
  2. If it is, what is the associated eigenvalue \(\lambda\)?
  
  
• Wednesday, Apr 26
  ¤ Two interacting populations of hares and foxes can be modeled by the discrete dynamical system \[ \vec{v}_{k+1} = A\cdot \vec{v}_k \quad \text{where} \quad A = \begin{bmatrix} 4 & -2 \\ 1 & 1 \end{bmatrix}. \] Find closed form solutions (Hint: Use eigenvalues and eigenvectors!) in the following three cases:
  
  1. \( \vec{v}_0 = \begin{bmatrix} h_0 \\ f_0 \end{bmatrix} = \begin{bmatrix} 100 \\ 100 \end{bmatrix} \),
     
     
  2. \( \vec{v}_0 = \begin{bmatrix} h_0 \\ f_0 \end{bmatrix} = \begin{bmatrix} 200 \\ 100 \end{bmatrix} \),
     
     
  3. \( \vec{v}_0 = \begin{bmatrix} h_0 \\ f_0 \end{bmatrix} = \begin{bmatrix} 600 \\ 500 \end{bmatrix} \);
  determine the long term values of \(\vec{v}_k\) in each case.
  ◊ Read "Do Hares Eat Lynx?" Michael E. Gilpin. American Naturalist, Volume 107, #957 (1973), pp 727-730.
  
  
• Friday, Apr 28
  ¡Quiz today!
  &loz Study: Test 1 Topics
  ◊ Bring questions from the Supplementary Exercises for Chapter 1 (pg 88)
  
  

Week 15    ♞  (This Week)

• Monday, May 1
  ◊ Study: Test 2 Topics
  ◊ Bring questions from the Supplementary Exercises for Chapters 2 & 3 (pg 185)
  
  
• Wednesday, May 3
  ◊ Study: Test 3 Topics
  
  
• Friday, May 5
  ¤ Final Exam from \( \fbox{Noon - 2:30 PM} \)
  
  ¤ The final is your last opportunity to demonstrate your excellence!
  • You can bring any notes — but no books!
  • You can use a calculator — check the batteries!
  • Bring a spare pencil/pen/eraser.
  
  ¤ Study:
  • Chapter 1 Supplemental Problems: pg 88
  • Chapter 2 Supplemental Problems: pg 160
  • Chapter 3 Supplemental Problems: pg 185
  • Chapter 4 Supplemental Problems: pg 262
  • Chapter 5 Supplemental Problems: pg 326
  
  ¤ Review sheets and practice exams from the text's authors.
  
  

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“On two occasions I have been asked, ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?’ I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.”

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