As part of an ILI grant from the NSF to Saint Louis University, we developed material to extend some of the teaching methods of calculus reform to upper level courses. One of the sub-project is to teach linear algebra in a computer classroom,using the computer whenever it seems appropriate for the material to be taught. We are using the computer algebra system Maple as the base tool on the computer.

Depending on what the instructor thought most effective for a particular class and section of material, the worksheets have been designed in several different models. I will describe them as the classroom activity model, the teacher demonstration model, and the Maple feature demonstration model.

The classroom activity model for worksheets -

• Assumptions:
  1. Organized classroom computer activities have the students working, in groups or independently, on computers with the instructor wandering around the room helping students when they get stuck and keeping them on task.
  2. The standard syllabus is so packed that computer activities need to be designed so that doing the activities covers new material from the syllabus rather than simply being practice exercises;
  3. The course is designed to teach mathematics rather than computer skills. Materials should be designed so the CAS learning curve does not interfere with the mathematics learning curve.
  4. When complicated coding is useful for a demonstration it should be in a format where the student can run it without being able to reproduce the programming. Parts of the code that the student will need to modify to look at different examples should be clearly labeled and easy to find.
  5. Exercises should require the student to explane or interpret what the computer is doing. (Raw output should generally not be an acceptable answer.)
• Implications:
  1. The worksheets contain enough text to be mathematically self explanitory;
  2. The worksheets have examples that will execute without modification;
  3. The worksheets have exercises where the student is required to consider other examples and interpret the results of computer calculations. The exercises have been designed so that the Maple programming skills needed are reduced to cutting, pasting, and modifying examples from the same worksheet;
  4. The worksheets can either be done in class or assigned as out of class assignments.

Classroom Activity worksheets for Linear Algebra I

These worksheets follow, section by section, the material in the text "Linear Algebra, Ideas and Applications" by Richard Penney, published by Wiley Press. The worksheets vary from, a straightforward adaptation of the material in the "On Line" sections of the book to use Maple and a worksheet model, to worksheets that develope supplemental material that the instructor thought appropriate at that point in the course.

• Just Enough Maple for Linear Algebra - This is a preliminary worksheet designed to introduce the students to the mechanics of working with Maple worksheets.
• Section 1.2 - Working with and plotting vectors The basic objectives are:
  1. Learn the basic mechanics of entering vectors as lists, and producing linear combinations with either addition or scalar multiplication.
  2. Learn to plot a set of vectors in R^2 and R^3.
  3. Using a random number generator, see what typical linear combinations of a pair of vectors look like.
• Section 1.3 - Systems of Linear Equations The basic objectives are:
  1. Use Maple to visualize the solution to a system of linear equations in R^2 .
  2. Use Maple to visualize the solution to a system of linear equations in R^3.
  3. Express the general solution in parametric form.
• Section 1.4 - Gaussian Elimination The basic objectives are:
  1. Converting between systems of equations and matrices
  2. Using Maple commands for elementary row operations
  3. Using more general Maple commands on matrices.
• Section 1.5 - Solving Matrix Equations The basic objectives are:
  1. Learn to use Maple to multiply matrices and vectors.
  2. Learn to use Maple to solve Matrix-vector equations.
  3. To do an extended exercise.
• Section 2.1 - Testing For Linear Independence The basic objectives are:
  1. Use Gauss-Jordan elimination to test the linear independence of column vectors of a matrix.
  2. Show that a set of real valued functions are linearly independent.
  3. Show that a set of real valued functions are linearly dependent.
• Section 2.2 - Dimension The basic objectives are:
  1. Use Gauss-Jordan elimination to find a basis for the column space of a matrix
  2. Use Gauss-Jordan elimination to find a basis for the null space of a matrix
  3. Note a relationship between the dimensions of these two spaces.
• Section 2.3 - Constructing Random Matrices with specified rank The basic objectives are:
  1. Learn how to produce a random matrix of specified size and rank
  2. Explore how Maple finds a basis for the row space and column space of a matrix.
  3. Relate the size of a matrix to the dimensions of the null space, the column space, and the row space.
• OnLine 3.1 - Visualizing Linear Transformations in R^2 The basic objectives are:
  1. Learn how to get Maple to draw simple stick figures in R^2, and how to do simple translations on the figures.
  2. Learn to represent a linear transformation in R^2 as multiplication by a matrix. Visualizing Linear Transformations in R^3;
• OnLine 3.2 - Visualizing Linear Transformations in R^3 The basic objectives are:
  1. Learn how to have Maple draw a stick figure of a car in R^3.
  2. Learn to represent a linear transformation in R^3 as multiplication by a matrix.
  3. Explore the relationships between the matrix of transformation and the image and nullspace.
• OnLine 3.3 - Image of a Transformation The basic objectives are:
  1. Investigate the image of a particular linear transformation
  2. Graphically investigate the null space of the linear transformation
• OnLine 3.5 - The LU Decomposition The basic objectives are:
  1. Compute the LU decomposition, if possible.
  2. See how elementary matrices are involved in the process.
  3. Understand how the LU decomposition is useful in solving systems of equations.
• OnLine 4.1 - Coordinates The basic objectives are:
  1. Visualize the notion of coordinates.
  2. Computing coordinates using the "coordinate matrix".
  3. Computing coordinates relative to an orthogonal basis.
• Online 4.3 - Fourier Approximations The basic objectives are:
  1. 1) Learn to compute Fourier approximations of functions on the closed interval [-1, 1].
  2. 2) Plot functions against their Fourier approximations to see that they are in fact good apporoximations.
  3. 3) See that Fourier approximations are effective for functions that do not look like they would be easily approximated by trigonometric polynomials.  
• Online 6.1 - Eigenvectors and Eigenvalues Goals of this worksheet:
  1. Compute eigenvalues and eigenvectors for several 3 x 3 matrices
  2. Use eigenvectors and eigenvalues to simplify a matrix computation.
• Online 6.2 - Diagonalization Goals of this worksheet:
  1. Construct a diagonalizable matrix which has specified eigenvalues and eigenvectors.
  2. Compute powers of diagonalizable matrices with ease!

• Assumptions:
  1. A CAS package like Maple can be used effectively to suppliment a traditional lecture by allowing the instructor to work with "non-rigged" examples that would otherwise be avoided because the details of the computations would obscure the point of the computations.
  2. An example worked with Maple can make the class more interactive by easily allowing exploration of similar problems. (The instructor can answer questions equivalent to "Does it always work like that, or is that feature of the result specific to this one example?")
  3. There are many mathematical points that are made clearer with the correct picture, but producing and justifying pictures by hand is hard work even for the artistically inclined.
• Implications:
  1. The worksheets contain enough text for the student to follow what is going on. The amount of text should be roughly comparable to what would be written on the board.
  2. The entire worksheet will execute without modification;
  3. The worksheet contains no exercises

Teacher Demonstration worksheets for Linear Algebra I

• Basic matrix computations - starts with a random matrix and walks through the computations needed to:
  1. determine if a vector is in the row space;
  2. show how the reduced row echelon form makes questions on membership in the row space easy to answer;
  3. examine the null space.
• Inverses - walks through the development of the process of finding the inverse of a matrix by solving a general system of equations with that matrix as the matrix of coefficients.
• Fourier approximations - walks through the computations needed to justify that the functions used in a Fourier polynomial are orthogonal and that they produce a reasonable approximation for an example function.
• Quadratic forms I - steps through the computations needed to rotate the axes of quadratic surfaces in 2 and 3 dimensions to remove the cross terms so the equations are in recognizable form.
• Quadratic forms II - repeats the process with another example in 3 dimensions.

• Assumptions:
  1. Besides structured activities, it is also useful for students to use a tool like Maple on there own.
  2. It is effective to have a simple worked example to show how they can use simple commands for independent work.
• Implications:
  1. These worksheets are short "ho to use Maple" worksheets.
  2. The worksheets will focus on simple commands rather than on exercises that require complicated coding.

Maple feature demonstration worksheets for Linear Algebra I

• Sec1-1-PlotLinSys.mws - Shows how to plot systems of linear equations in 2 and 3 variables.
• Sec1-2-GaussElim.mws - shows how to use Maple to solve a matrix equation by Gaussian elimination by using elementary operations.
• Sec2-2-BasicMatOps.mws - shows how to add matrices, multiply by scalars, transpose matrices, and extract entries of a matrix.
• Sec2-3-InvertMatrix.mws - shows how to use Maple to invert a matrix and to verify that the result is the inverse.

The second course we teach in Linear Algebra is more abstract focusing more heavily on proofs and appropriate inclusion of computer algebra is less clear.

Two worksheets were created for this class using the Maple feature demonstration model. One of them looked at creating matrices with specified Jordan and Rational Blocks, and finding Jordan and Rational bases for a matrix. This was intended to help the students work with nontrivial examples to be able to follow the proofs in the study of canonical forms. The second worksheet of this type looked at the Maple commands for doing Gram Schmidt orthogonalization.

A block of 3 worksheets was developed for the study of orthogonal polynomials as a case study of inner product spaces. It was assumed that the students had already seen inner product spces in the context of Fourier series. The worksheets have the students work through how different inner products produce a different definition of close and what that means in terms of a function space. The first worksheet compares Legandre Polynomial approximation against the more familist Taylor series approximation. The second worksheet compares Chebyshev Polynomial approximation against Legandre Polynomial approximation. The third worksheet worksheet compares Jacobi Polynomial approximations with different weight functions.

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