Report of the Technology Subgroup

Introduction

Index

1. Maximizing Strengths While Minimizing Weaknesses of Technology in the Classroom

   • Choosing to Utilize Technology
   • Some effective Uses
   • Possible Pitfalls

2. Examples of Lessons Enhanced by Techology

   • Fernanda Botellho
   • Jane M. Day
   • Paul Hurst
   • Michael Penkava
   • John Polking
     • Age-Based Matrix Population (Leslie) Models
     • Exercises
     • Stage-based models: Application to Loggerhead sea turtles
     • Exercises
   • Lorenzo Sadun
   • John Wicks

   • References

3. Workshop Materials

   Demonstration Exercises
   • MapleV release 5
   • Mathematica 3.0
   • Matlab 5
   • TI-86
   • TI-92

4. List of Participants

Despite these differences, we all agreed on one over-arching principle. Regardless of whether one will be creating new materials or using materials prepared by someone else, one should have a clear educational objective, and consider whether the use of technology will enhance this purpose enough to outweigh the time and effort required to implement it. If there is no compelling advantage to using it, technology should probably be avoided.

Technology can also introduce new and unanticipated types of difficulties. Equipment and software will not always work properly, and there can be administrative hassles. Anyone considering using technology should determine what institutional sources of technical support will be available. It is strongly recommended that a graduate or undergraduate student who has experience with the technology be available in any computer lab setting (Note: This often eventually occurs on its own over time if the same technology is used consistently in a number of courses).

So at the next class I started MATLAB and the projector, stored the matrix as A, and then typed R = rref(A). They all agreed this was what they had done, and then they had inspected the row-reduced echelon form to identify four pivot columns in A, put those four columns into a matrix B, and calculated rref([B y]) for one of the two nonpivot columns y. They seemed confident about this, but still were interested to learn that everyone had solved the problem the same way. Then I told them they didn't really need to do the second calculation – the solutions to Bx = y were already visible in the row-reduced echelon form of A, for both nonpivot columns y. Apparently no one had noticed that, but they stared at A and R as I covered up columns appropriately, and saw quickly that each nonpivot column of R is a linear combination of the pivot columns on its left; and then saw why that same relationship is valid for the columns of A. I pointed out this would be true for any matrix, for the same reasons. They seemed very satisfied with this discussion. In the past I've done similar discussions using a chalkboard, but the computer demonstration was definitely more vivid and held their keen attention.

Let A be a symmetric n by n matrix whose eigenvalues in decreasing size order range from l₁ to l_n. If x is an n by 1 vector whose euclidean norm is one, then:

1. The inner product of Ax and x is between l₁ and l_n.

2. The inner product of Ax and x is equal to l₁ if x is an eigenvector of A corresponding to l₁ and the inner product of Ax and x is equal to l_n if x is an eigenvector of A corresponding to l_n.

> with(linalg):
> A := matrix(2,2,[1, 2,2,1]);eigenvalues(A);

The following short program will then compute a range of values for x^TAx, as x ranges around the unit circle:

> for r from 0 to 10 do
> t := evalf(2.0*Pi*r/10):
> x:=matrix(2,1,[cos(t), sin(t)]):
> v[r] := evalm(transpose(x)&*A&*x);
> print(t,evalm(v[r]));
> od:

We can then define a function f1(t) which represents x^TAx:

> f1 := proc(t,A)
> local x;
> x := vector([cos(t),sin(t)]);
> innerprod(x,A,x);
> end;

> l := [[ 2*Pi*n/100, f1(2*Pi*n/100,A)] $n=1..100]:
> plot(l, x=0..2*Pi, style=point,symbol=circle);

> f2 := proc(t)
> local x;
> x := vector([cos(t),sin(t)]);
> innerprod(x,A,x);
> end;
> plot(f2(t),t=0..6);

Problem 1: Let S be the subspace of R⁴ spanned by the vectors u₁:=(1,2,3,4), u₂:=(4,2,1,5), and u₃:=(3,5,1,7). Determine if the vectors v:=(8,9,5,16) and w:=(7,2,1,3) are in S.

This problem is solved by considering the expression v:= x₁u₁ + x₂u₂+x₃u₃ as a system of 4 equations in 3 unknowns. In order to solve this problem using MapleV, the elements of R⁴ are entered as vectors. Thus, the equation for v above is really a vector equation, rather than a system of equations. The student will need to learn how to convert the vector expression into a system of linear equations, and the textbook gives a Maple procedure which will take care of this conversion. Solving this system directly yields the solution {x₁=1, x₂=1,x₃=1}. A similar approach with w will yield no solution. An alternative method is to consider the augmented matrix A:=[u₁ u₂ u₃ v w], and find its reduced row echelon form. The latter method also allows one to express v as a linear combination of the vectors u₁, u₂, and u₃, by looking at the coefficients appearing in the fourth column of the reduced row echelon form.

This exercise on Matrix Population Models should actually be considered a project. It is too long and difficult to be a simple exercise. It demonstrates an actual use of matrices and their eigenvalus and eigenvectors. It would be very hard to complete the project without using a computer, since it requires the computation of eigenvalues and eigenvectors for many matrices which are four by four and five by five. Furthermore, the entries are not small rational numbers, and in the more interesting second section the entries themselves need to be computed. The project is best done by doing some elementary programming. It can be done without that, but it is difficult, especially in the second section. The needed programming can easily be done in MATLAB, and probably is not difficult in other mathematical programs. The project encourages the student to experiment with the model to see the effects of different ecological strategies. It should be emphasized that the model in the second section was actually used by the federal government and by some state governments to decide on their ecological strategies.

In this exercise we will illustrate the application of eigenvalues and eigenvectors to the analysis of models of population growth. In the final part we will employ actual data about the life cycles of loggerhead sea turtles to make predictions regarding their survival. We will also use the model to analyze the effectiveness of several strategies for improving the chance that the species will survive.

There are two factors involved — birth and death — that change the populations from year to year. We will assume that each age group has its own death rate d_k, and define the survival rate for this group by s_k = 1 - d_k. Thus is the number of newborns in cycle n + 1 and will be the sum of the births to all age groups. The group of age k - 1 must first survive to age k and then reproduce, so the number of births from this group will be . The total number of newborns in cycle n + 1 will therefore be

(2)

. (3)

We can analyze the evolution of the population by using information about the eigenvalues and eigenvectors of the projection matrix. For now, let's assume that R has A distinct eigenvalues, r₁, ..., r_A. For each of the eigenvalues, r_k, we can find an associated nonzero eigenvector v_k. Under the assumption that the eigenvalues are distinct, it follows that this set of eigenvectors form a basis. Once again, this means that the initial population vector p⁽⁰⁾ can be written as a linear combination of the eigenvectors, i.e., there are constants c_k such that

. (6)

(7)

Before proceeding further, we will make another hypothesis. We will assume that one of the eigenvalues is real, positive, and larger than the absolute values of each of the others. We will assume that we have labeled the eigenvalues so that this one large eigenvalue is r₁. r₁is called the dominant eigenvalue. (MATLAB may not choose to place the dominant eigenvalue first when calculating eigenvalues.)

. (8)

p⁽ⁿ⁾ (9)

.

• We will work with an age-based model first. Consider a population which reproduces at the end of each of the years of life, 1, 2, 3, and 4, with birth rates 4/5, 3/5, 2/5, and 1/5. Suppose the survival rates for these four years are 1, 3/4, 1/2, and 1/4.
  1. Set up the population projection matrix R.

  2. Compute the eigenvalues of R and show that there is a dominant eigenvalue r₁. (Two of the eigenvalues will be complex but neither has the dominant magnitude.) Normalize the associated eigenvector v₁ so that the sum of the terms is equal to 1. Do you predict that the species will flourish or perish? What will be the asymptotic ratio of populations in these four years of life?

  3. Now write a program to calculate the exact population dynamics for this model by calculating p⁽ⁿ⁾ from Rⁿp⁽⁰⁾, as a function for n for 0 £ n £ 20. You can take as the initial conditions p⁽⁰⁾ = [1, 0, 0, 0]^T. According to equation (9), it is easier to see the asymptotic behavior of p⁽ⁿ⁾ by plotting versus n. By "plot," we mean plot each component versus n for 1 £ k £ 4. Demonstrate that the asymptotic limit predicted in equation (9) is indeed observed.

   

• What happens if there is no dominant eigenvalue? As a slight variation on the model in Exercise 1, consider a population which reproduces only after the second and fourth years with birth rates 2/3 and 1/3. Suppose the survival rates for these four years are 1, 3/4, 1/2, and 1/4.
  1. Set up the population projection matrix R.

  2. Compute the eigenvalues and eigenvectors, and show that there is no dominant eigenvalue.

  3. Using your program from Exercise 1c, compute and plot for 0 £ n £ 20, where p⁽⁰⁾ = [1, 0, 0, 0]^T. The limiting phenomenon in this case is called a population wave. Write a paragraph describing the limiting behavior and how it differs from that of the previous example.

 

Stage-based models: Application to Loggerhead sea turtles

In the last thirty years, the age-based models described in the previous section have been generalized to what are called stage-based models. In this case the population is separated into a number of "stages," which may or may not depend on age. Again we have a population vector p, with components p_K representing the number of individuals in the k^th stage.

 

As before, we assume that the species undergoes a cycle of reproduction and change. During this cycle the individuals in the k^th stage may stay in the k^th stage or be transformed into one of the other stages. The probability that an individual in the k^th stage will go into a new stage, i, is denoted by r_ik. Thus if p⁽⁰⁾ is the population vector in the n^thcycle, the number of individuals in the k^th stage at the (n + 1)^st cycle is

If we put the stage change probabilities into a matrix R, then the transition from cycle n to cycle n + 1 is given by the same matrix equation

p^{(n + 1)} = Rp⁽ⁿ⁾,

just as it was for the models in the previous section. Consequently, we have the analog of equation (5)

p⁽ⁿ⁾ = Rⁿp⁽⁰⁾ (10)

We will again call the matrix R the population projection matrix.

 

Since everything in the previous section depended on (5), all of the results of that section apply here as well. In particular this is true of the analysis of the asymptotic behavior of the population vector, and the importance of a dominant eigenvalue and its associated eigenvector.

 

We will now apply these concepts and methods to the population of loggerhead sea turtles. Marine turtles, as typified by the loggerheads are long-lived and mobile³. Only the adult nesting females, eggs and hatchlings, and stranded, dying turtles are ever seen on beaches. Furthermore, no method has been devised to obtain accurate age information about sea turtles. These reasons pretty much preclude obtaining information of the detail that would be required to use an age-based model of the type described in the previous section. For these reasons, it was decided to use a stage-based model, by dividing the population into five stages. The stages, stage durations, survivorships, and birth rates are shown in Table 1. The data in Table 1 are the result of long observation of loggerhead sea turtles⁶.

 

Table 1. Five-stage life history parameters for loggerhead sea turtles.

 

 

An individual in one of the stages can survive and stay in the same stage or it can survive and move on to the next stage. Consequently the population projection matrix has the form

(11)

where G_k is the probability of an individual at stage k surviving and advancing to stage k + 1, P_k is the probability of an individual surviving and remaining in the same stage, and F_k is the reproductive rate of the k^th stage.

 

Calculating the probabilities in the population projection matrix for the stages that have a duration longer than one year is a little complicated, so we will go through it step by step. If the duration is d years in stage k, then there are individuals in that stage who have been there for 1, 2, …, d years. The probability of any individual surviving to the next year is the survival rate, s _k, in Table 1. The probability of surviving y years is . Consequently the number of individuals in the various age groups relative to the number of age 1 is 1, , , …, . Hence the total number of individuals in the k^th stage, relative to the number of age 1, is

 

Only the individuals with the highest age in a stage move onto the next stage. Hence, the relative number of them is , and the probability of an individual in the k^th stage growing into the next stage is the ratio

(12)

(13)

Notice that when d = 1,

G_k = g_k, (14)

P_k = (1 - g_k) (15)

• Write a program (i.e., a MATLAB function M-file) that inputs the vector of survival rates and outputs the projection matrix. As a check on your program, the answer for the data in Table 1 is
  

• Assuming this model is correct, describe what will happen to the population of loggerhead sea turtles as time passes.

• Prior to the mid 1980s, almost all conservation dollars spent on the preservation of loggerhead sea turtles was spent to improve the survivability of eggs and hatchlings (i.e., stage 1). To see the maximum effect that this policy can attain, set s₁ = 0.999 (i.e., 99.9% survivability), and recompute R. Describe the asymptotic behavior of the population with this new data.

• Let's assume that TEDs have the effect of reducing the death rates (i.e., 1 - s_k) of turtles in stages 3, 4, and 5 by the same amount. I.e., if the reduction is p, then we are replacing s_k with s_k + p(1 - s_k)$ for i = 3, 4, 5. Write an M-file that computes the dominant eigenvalue of the projection matrix for values of p between 0 and 0.5. Plot the eigenvalue against p. Describe your results and the implication about the use of TEDs. Compare these results with those of exercise 5.

1. Using technology, compute PA, AP, EA, AE, TA, AT, DA, and AD.

2. How do PA, EA, TA, and DA compare to A? Describe in words what multiplying on the left by P, E, T and D does to a matrix.

3. Would the answer to (b) be any different if A were not a square matrix? Try typing in a rectangular matrix B (say, 5 by 7), compute PB, EB, etc. and describe the results.

4. Let v by a 5 by 1 matrix, in other words a column vector. What do the matrices P, E, T, and D do to v ?

5. How do AP, AE, AT, and AD compare to A? If C is a 7 by 5 matrix, how do CP, CE, etc. compare to C? Describe in words what multiplying on the right by P, E, T, and D does to a matrix.

6. Let , let G be a general 2 by n matrix, and let H be a general n by 2 matrix. Without doing any computations, describe FG and HF in terms of G and H. Then play with specific examples of matrices G and H and see if your description is correct.

1. Anton, Howard, Elementary Linear Algebra, 7^th Edition, John Wiley and Sons, Inc., 1994.

2. Bauldry, Evans, and Johnson, Linear Algebra with Maple, John Wiley & Sons, 1995.

3. Crouse, Crowder, and Caswell, A Stage-based Population Model for Loggerhead Sea Turtles and Implications for Conservation, Ecology, 68(5), 1987, pp. 1412--1423, and Crouse, Crowder, Heppell, and Martin, Predicting the Impact of Turtle Excluder Devices on Loggerhead Sea Turtle Populations, Ecological Applications, 4(3), 1994, pp. 437 – 445.

4. Day, Jane M. , Instructor’s MATLAB Manual for Linear Algebra and its Applications, 2^nd Edition. Addison Wesley Longman, 1997.

5. DeAlba, Luz Maria, Instructor’s TI-85 Manual for Linear Algebra and its Applications, 2^nd Edition. Addison Wesley Longman, 1997.

6. Frazier, N. B., Survivorship of adult female loggerhead sea turtles, Caretta caretta, nesting on Little Cumberland Island, Georgia, USA, Herpetologica 39, 1983, pp. 436 – 447.

7. Golubitsky and Dellnitz, Linear Algebra and Differential Equations using Matlab. Brooks Cole, 1999.

8. Herman, King, Moore, and Pepe, LAMP (Linear Algebra Maple Project). Addison-Wesley. To appear. http://www.math.grin.edu/lamp/

9. Hill and Zitarelli, Linear Algebra Labs with MATLAB, 2^nd Edition, Prentice Hall, 1996.

10. Johnson, Eugene, Linear Algebra with Maple V, Brooks/Cole, 1993.

11. Lay, David C., Instructor’s Maple Manual for Linear Algebra and its Applications, 2^nd Edition. Addison Wesley Longman, 1997.

12. Lay, David C., Linear Algebra and Its Applications, 2^nd ed., Addison Wesley Longman, 1997.

13. Leon, Herman, and Faulkenberry, ATLAST Computer Exercises for Linear Algebra. Prentice Hall, 1996.

14. Polking, John C., Ordinary Differential Equations using MATLAB, Prentice Hall, 1995.

15. Smith, Rick L., MATLAB Project Book for Linear Algebra. Prentice Hall, 1998.

16. Wicks, John R., Linear Algebra with Mathematica, An Interactive Approach. Addison Wesley Longman, 1996.

1. Solve the following system of equations:
   
   Here are several ways:
   1. Create the coefficient matrix and target vector by typing

      A = [1 2 2 5; 2 1 5 –2; 1 1 3 1]
      b = [-10; 26; 8]
      rref([A b])

       

   2. You can type several commands on one line if you wish, by putting commas between them. For example, you could accomplish the above by typing

      A = [1 2 2 5; 2 1 5 –2; 1 1 3 1], b = [-10; 26; 8], rref([A b])

      or by creating the augmented matrix to start with, and reducing that:

      M = [1 2 2 5 -10; 2 1 5 –2 26; 1 1 3 1 8], rref(M)

    

2. The Suburb-City problem. Type A = [0.6 0.2; 0.4, 0.8] to store the transition matrix. Let p0 denote the initial vector.

    

   1. Type p0 = [5; 4], p = p0 to store the first value given for p0, and make a copy of that.

   2. Type p = A*p to calculate the population vector after one time period. Press the up arrow ­ on keyboard to retrieve the line p = A*p and press [Enter] to execute it again. Repeat as many times as you want, to see the series of iterates converge.

   3. To plot the values of several iterations, try typing this:

      p = p0; P = [p0]; for i = 1:12, p = A*p; P = [P p]; end

      (i.e., store p0 and 12 iterates in P, using ";" to suppress printing) P (see P)

      x = 2000:2013; plot(x, P)

      (Store x-values in x and plot each row of P against x)

      axis([2000 2015 5 12] , shf

      (Set limits for x and y axes and type shf to show figure)

   4. Find an equilibrium vector by using eigenvalues of A.
      One way is to directly solve p = Ap, i.e., (A – I)p = 0, by typing

      rref([A – eye(2)]

      and writing the general solution. Another way is to calculate all eigenvalues and eigenvectors of A and select an eigenvector corresponding to 1. To do that, type [V D] = eig(A). You will see

      

       

      

       

      The diagonal entries of D are the eigenvalues of A, and column i of V is an eigenvector corresponding to D(i,i). So column 2 of V is an eigenvector corresponding to the eigenvalue 1. To store that as a separate vector, type p = V(:, 2) .

       

      The algorithm in eig always produces columns in V which have Euclidean norm 1, but it is easy to scale these as you wish. For example, to get an eigenvector corresponding to the eigenvalue 1 which has sum 9, type p = (9/sum(p))*p.

    

3. To repeat the Suburb-City example with S₀ = 9 and C₀ = 3, typ p0 =[ 9; 3], p = p0 and then repeat the calculations in Exercise 2 (b) – (d). The easy way is to use the up arrow to retrieve each of the lines typed before (one line at a time) and execute them again.

1. 2^nd Matrix
2. Edit (F2)
3. Input any letter that you want to name the matrix
4. Enter the dimension of the matrix
5. Input the entries
6. Exit
7. Check the entries by inputting the letter you have chosen to name the matrix and pressing enter.

1. The first step is to input the augmented 3 ´ 5 matrix that corresponds to the first exercise. Once this is done, proceed with the following steps.
   1. 2^nd Matrix
   2. Ops (F4)
   3. Rref (F5) (Can you guess what rref stands for?)
   4. Enter the name of the matrix
   5. Enter
   Now, what you should have is the reduced row echelon form for the augmented matrix. Translate back to a system of equations, and you know what to do from there.

    

2. Use the matrix editing steps to input the 2 ´ 2 matrix A and then the 2 ´ 1 matrix V with entries 5 and 4. For part a), do the following steps.
   1. A*V Enter
   2. A*2^nd Ans Enter
   3. 2^nd Entry Enter
   4. Repeat step 3 as many times as you like.
   As another way of investigating the problem, try entering A^100*V.
   For part b), you can get the information from the work for part a). Finding the eigenvalues and eigenvectors of A on the TI-86 doesn’t work as well. Here are the steps to do it.
   1. 2^nd Matrix
   2. Math (F3)
   3. EigVl (F4) or eigVc (F5)
   Note that the Eigenvalues are fine, but the eigenvectors are perhaps not in the form you would like. I guess the default is to write them as column vectors of norm 1. We don’t usually teach the students how to find the eigenvectors in this way. A more informative method would be to find the eigenvalues c, and then find the reduced row echelon form of A - cI.
   For part c), just input a 2 ´ 1 matrix with entries 3 and 9, and proceed as in part b.

    

3. Let’s find the determinant and inverse of D =
   For the determinant:
   1. Input the matrix D as above
   2. 2^nd Matrix
   3. Math (F3)
   4. Det (F1)
   For the inverse:
   1. D 2^nd x^-1 (EE key). Now, to put in fractional form
   2. 2^nd Matrix
   3. Misc (F5)
   4. More
   5. Frac (F1)
   The steps to change into a fraction are probably long so naïve students won’t hurt themselves.
   Advantages: The calculator is much easier to carry around, takes up less space, and is cheaper than a computer. Students can carry them to exams, bring them to class and use them to do homework much more conveniently than they could use a computer. You also don’t have to wait for it to boot up and shut down.
   We use the graphing calculator to help our college algebra students do linear algebra. They use them to do Gaussian elimination, and to find inverses and determinants of matrices. With the help of the calculator, they find these things relatively easy to do.
   Disadvantages: The calculator has limited memory space. It requires more effort to save your work and print things out. It has limited ability to do symbolic manipulation.

1. Press the Apps key
2. Go down to the Data/Matrix Editor and across to new
3. Give the matrix a name in the variable slot
4. Use the right and down arrow to change data to matrix
5. Edit the row dimension and column dimension and press enter
6. Input the entries in the table
7. Exit by pressing 2^nd and Quit

1. The first step is to input the augmented 3 ´ 5 matrix that corresponds to the first exercise. Once this is done, proceed with the following steps.
   1. 2^nd Math
   2. Scroll down to matrix, and across to rref( and then press enter
   3. Enter the name of the matrix, close off the parentheses and press enter
   Now, what you should have is the reduced row echelon form for the augmented matrix. Translate back to a system of equations, and you know what to do from there.

2. Use the matrix editing steps to input the 2 ´ 2 matrix A and then the 2 ´ 1 matrix V with entries 5 and 4. In doing this, you may need to edit a matrix you have already entered. You can delete rows or columns by using Util (F6) Once you have put in the matrices, do the following.
   1. A*V Enter
   2. A*2^nd Ans Enter
   3. Enter
   4. Repeat step 3 as many times as you like.
   As another way of investigating the problem, try entering A^100*V.
   For part b, you can get the information from the work for part a. Finding the eigenvalues and eigenvectors of A on the TI-92 is very different. Here is one possible outline of the steps.
   1. Find the characteristic equation (det(A-x))
   2. Find the roots of the characteristic equation
   3. Find the reduced row echelon form of A-x for each eigenvalue
   For part c, just input a 2 ´ 1 matrix with entries 3 and 9, and proceed as in part b.

3. Let's find the determinant and inverse of D =
   1. Input the matrix D as above
   2. 2^nd Math
   3. Go to matrix, and then over to det(
   4. Enter D, and close off the parentheses
   You can find the inverse by
   1. D 2^nd x^{-1} (9 key)
   Notice that the answer is in fractional form. If you want decimal form, press the green diamond button followed by the green wavy equals sign.
   The TI-92 can do symbolic manipulation, it has pull-down menus and the display screen is larger. It is larger than the TI-86 and costs more. In general, it also has all of the advantages and most of the disadvantages of the TI-86 mentioned in the previous section.