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The Scientific Notebook

Activities

1. Matlab and Vectors
2. Matlab and Matrices
3. Reduced Row Echelon Form. Matlab's rref Command
4. Numerical Linear Algebra.
5. Writing Scientific Papers in LaTeX.
6. Interpolating Polynomials
7. Population Modeling Using the Leslie Matrix, Part 1
8. Matlab and Coordinate Systems
9. Linear Versus Non-Linear Transformations
10. Least Squares Approximation
11. Composition and Inverses of Transformations
12. Graphs and Matlab
13. Digraphs and Matlab
14. Modular Arithmetic and Hill Ciphers in Matlab
15. Deciphering Hill Ciphers
16. Matrix Multiplication - Alternate Methods
17. Properties of Determinants and Matlab
18. The Dimension of Vector Spaces
19. Difference Equations - Closed Form Solutions
20. Dynamical Systems - Long Term Behavior
21. Population Modeling Using the Leslie Matrix, Part 2
22. Stage-Based Population Model - Loggerhead Sea Turtles
23. The Projection Matrix
24. The Projection Matrix Revisited

Matlab and Vectors

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Abstract-

Prerequisites-

• none

Matlab and Matrices

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Abstract-

Prerequisites-

• none

Reduced Row Echelon Form. Matlab's rref Command.

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Abstract-

Prerequisites-

• Some familiarity with pencil and paper calculations required to place a matrix in reduced row echelon form.

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Sys34.m (A Matlab M-file used in the activity)

Numerical Linear Algebra

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Clicking on the following links will open the pdf file in your browser, provided you have the appropriate plugin. Normally, when you install the Acrobat Reader, the appropriate plugins for your browser are automatically installed. A second option is to right-click each link, saving both screen version and print version to your hard disk. They must be saved in the same directory. Open the screen version in the Acrobat Reader. Once open, select Full Screen in the Acrobat Reader for optimal viewing.

Abstract-

In this activity you will learn how to solve systems of linear equations using LU decomposition, with both forward and back substitution. In addition, you will also be introduced to function files and a number of \mat's control structures.

Prerequisites-

• Some knowledge of how to enter vectors and matrices in Matlab. Familiarity with Gaussian elimination, elementary matrices, and LU decomposition.

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Writing Scientific Papers in LaTeX

Download Instructions

Clicking on the following links will open the pdf file in your browser, provided you have the appropriate plugin. Normally, when you install the Acrobat Reader, the appropriate plugins for your browser are automatically installed. A second option is to right-click each link, saving both screen version and print version to your hard disk. They must be saved in the same directory. Open the screen version in the Acrobat Reader. Once open, select Full Screen in the Acrobat Reader for optimal viewing.

Abstract-

In this activity you will learn how to typset scientific papers in LaTeX. This is a preparation for your efforts on the term project.

Prerequisites-

• You need a TeX installation on your computer.

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Interpolating Polynomials

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Clicking on the following links will open the pdf file in your browser, provided you have the appropriate plugin. Normally, when you install the Acrobat Reader, the appropriate plugins for your browser are automatically installed. A second option is to right-click each link, saving both screen version and print version to your hard disk. They must be saved in the same directory. Open the screen version in the Acrobat Reader. Once open, select Full Screen in the Acrobat Reader for optimal viewing.

Abstract-

A polynomial that passes through a given set of data points is called an interpolating polynomial. In this exercise you will use Matlab to find the interpolating polynomial (if one exists) for a variety of different data sets. The Vandermonde matrix is introduced as an efficient means of entering the augmented matrix.

Prerequisites-

• Solving linear systems with Matlab's rref command.
 

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Population Modeling Using the Leslie Matrix, Part 1

Download Instructions

Clicking on the following links will open the pdf file in your browser, provided you have the appropriate plugin. Normally, when you install the Acrobat Reader, the appropriate plugins for your browser are automatically installed. A second option is to right-click each link, saving both screen version and print version to your hard disk. They must be saved in the same directory. Open the screen version in the Acrobat Reader. Once open, select Full Screen in the Acrobat Reader for optimal viewing.

Abstract-

The Leslie Model is a powerful tool used to determine the growth of a population as well as the age distribution within a population over time.

Prerequisites-

• Matrix multiplication
• Some familiarity with Matlab's indexing methods.
 

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Matlab and Coordinate Vectors

Download Instructions

Clicking on the following links will open the pdf file in your browser, provided you have the appropriate plugin. Normally, when you install the Acrobat Reader, the appropriate plugins for your browser are automatically installed. A second option is to right-click each link, saving both screen version and print version to your hard disk. They must be saved in the same directory. Open the screen version in the Acrobat Reader. Once open, select Full Screen in the Acrobat Reader for optimal viewing.

Abstract-

In this exercise we will introduce the concept of a coordinate system for a vector space. The map from the vector space to its coordinate space is defined, then graph paper is crafted to accompany the corresponding basis. Moving from one coordinate system to another is explained, and the activity uses different kinds of graph paper to help the reader visualize the change of basis technique.
 

Prequisites:

• Knowledge of a basis for a vector space.
 

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Linear Versus Non-linear Transformations

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Clicking on the following links will open the pdf file in your browser, provided you have the appropriate plugin. Normally, when you install the Acrobat Reader, the appropriate plugins for your browser are automatically installed. A second option is to right-click each link, saving both screen version and print version to your hard disk. They must be saved in the same directory. Open the screen version in the Acrobat Reader. Once open, select Full Screen in the Acrobat Reader for optimal viewing.

Abstract-

Linear transformations must map lines to lines (or points). This is not the case with non-linear transformations. this Matlab activity will encourage visual exploration of this concept.

Prerequisites-

• Matrix multiplication, rudimentary knowledge of the nullspace, dependent and independent vectors, bases, linear transformations, Matlab's elementwise operators.
 

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Least Squares Approximation

Download Instructions

Clicking on the following links will open the pdf file in your browser, provided you have the appropriate plugin. Normally, when you install the Acrobat Reader, the appropriate plugins for your browser are automatically installed. A second option is to right-click each link, saving both screen version and print version to your hard disk. They must be saved in the same directory. Open the screen version in the Acrobat Reader. Once open, select Full Screen in the Acrobat Reader for optimal viewing.

Abstract-

Scientists and mathematicians often have need to ``fit'' a line to a set of data points. In this activity, you will learn how linear algebra greatly simplifies this task.

Prerequisites-

• Matrix operations, transpose and inverse, knowledge of the column space of a matrix.
 

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Composition and Inverses of Transformations

Abstract-

An activity (actually a homework assignment as much of this material was demonstrated during class) designed to interpret geometrically the composition and inverses of transformations.

Prerequisites-

• Two Matlab 5 M-files, gtrans.m and cmptrans.m and their accompanying callback routines, gtr.m and cmptr.m
 

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Graphs and Matlab

Abstract-

Graphs are introduced along with their adjacency matrix. Paths are defined and an algorithm posed for counting paths of length n.

Prerequisites-

• Matrix multiplication
• A little knowledge of the transpose of a matrix
• You will also need to download and use the Matlab file graph.m.
 

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Digraphs and Matlab

Abstract-

Digraphs are introduced along with their adjacency matrix. An algorithm is proposed for counting paths of length n.

Prerequisites-

• Matrix multiplication
• You also need to have completed the activity on graphs before proceeding to this activity on digraphs.
 

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Modular Arithmetic and Hill Ciphers in Matlab

Abstract-

A short but practical introduction to modular arithmetic is given. Matrix multiplication is used to encrypt plaintext.

Prerequisites-

• Matrix multiplication
 

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Deciphering Hill Ciphers

Abstract-

The purpose of this activity is to decipher ciphertext that has been encoded with Hill ciphers. A review of the inverse of a matrix is included and particular attention is paid to the inverse of a matrix modulo 26.

Prerequisites-

• Matrix multiplication
• The inverse of a matrix
• Also, the reader should first complete the activity "Modular Arithmetic and Hill Ciphers in Matlab" before attempting this activity.
 

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Matrix Multiplication - Alternate Methods

Abstract-

Matlab is used to demonstrate matrix multiplication by columns and rows.

Prerequisites-

• A basic understanding of ordinary matrix multiplication
 

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Properties of Determinants and Matlab

Abstract-

Matlab is used to investigate a number of important properties of the determinant. The effect of the elementary row operations on the determinant is examined and the determinants of triangular, identity, and permutation matrices are calculated. The LU decomposition is introduced as used to find the determinant of a matrix.

Prerequisites-

• Elementary row operations
• Rudimentary understanding of the determinant
 

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The Dimension of a Vector Space

Abstract-

If the set of vectors B={v₁,v₂,...,v,_n} are linearly independent and span the vector space V, then the set B is called a basis for the vector space V. The number of vectors in the set B is called the dimension of the vector space V. In this activity you will use a number of Matlab commands to explore and reinforce these ideas.

Prerequisites-

• Familiarity with Matlab's rref command and its use to solve linear systems is assumed. Readers should also be familiar with the concepts of linear independence and the span of a set of vectors.
 

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Difference Equations - Closed Form Solutions

Abstract-

This activity investigates the form of closed form solutions of first order difference equations of the form u_k = Au_{k - 1} where A is a 2 x 2 matrix.

Prerequisites-

• This activity requires a knowledge of eigenvalues and eigenvectors
 

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Dynamical Systems - Long Term Behavior

Abstract-

The closed form solution of difference equations is used to predict the long-term behavior of dynamical systems. The role of the "dominant" eigenvalue is introduced.

Prerequisites-

• Eigenvalues and eigenvectors
• You should first complete the activity Difference Equations - Closed Form Solutions before attempting this activity.
• You need to download a copy of traj2.m to complete this activity.
 

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Population Modeling Using the Leslie Matrix, Part 2.

Abstract-

The Leslie Model is a powerful tool used to determine the growth of a population as well as the age distribution within a population over time.

Prerequisites-

• Linear systems
• Matrix multiplication
• Eigenvalues
• Eigenvectors
• Diagonalization.
 

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Stage-Based Population Model - Loggerhead Sea Turtles

Abstract-

A stage-based population model is used to model population growth of loggerhead sea turtles. Model is based on data collected from Little Cumberland Island, Georgia.

Prerequisites-

• A working knowledge of the Leslie model of age-based population. An understanding of eigenvalues, eigenvectors, and the closed form solution of x_k = Ax_{k - 1} is also needed.
 

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The Projection Matrix

Abstract-

In this activity you will use Matlab to project a set of vectors onto a single vector.

Prerequisites-

• Inner product (dot product)
• orthogonal vectors
 

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The Projection Matrix Revisited

Abstract-

Matlab is used to demonstrate that vectors in the column space of A^T are orthogonal to vectors in the nullspace of A. A similar relationship is demonstrated for the column space of A and the nullspace of A^T. A geometric argument is used to find a matrix P that will project vectors from V onto a subspace W of V.

Prerequisites-

• Knowledge of the inner product (dot product) and orthogonality conditions are required. Before attempting this activity, definitions of the subspaces mentioned above should be completed and students also need a fair understanding of the bases for each of the mentioned spaces.
 

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