Matrix Algebra Demos

 
Objective: To provide movies with audio for basic matrix algebra computations that can be used by instructors as part of a lecture and by students for self-study.

Level: High school or college courses in which basic matrix algebra techniques are introduced.

Prerequisites: Familiarity with a matrix, its entries, and their row-column addresses is required. The animation on a the matrix vector product assumes familiarity with linear combinations. If the animation on the method of elimination is to be used, then the concepts of coefficient matrix, right side and augmented matrix will be needed. For the animations based on the MATLAB routine reduce, familiarity with row operations is required.

Platform: No particular software is required for the movies on matrix algebra computations, other than the Flash player, which is a free download. Go to http://www.adobe.com/support/flash/downloads.html

If you have MATLAB, the routine reduce, described below, can be downloaded by clicking here.

Routine reduce performs row reduction on matrix by explicitly choosing row operations to use. A row operation can be "undone", but this feature cannot be used in succession. This routine is for small matrices, real or complex.

Note: the routine reduce also works in the MATLAB look-alike program Octave which is a free download from http://www.gnu.org/software/octave/download.html

Instructor's Notes: The matrix algebra computations of Dot Product, Matrix Addition, Matrix-Vector Product, Matrix Multiplication, and the Method of Elimination are traditionally done with discussion and examples on a chalk board or white board. Here we present an alternative mode of presentation utilizing Flash movies with audio that present step-by-step examples of the operations or procedures. The movie can be stop and reversed to review or emphasize a point, or to respond to student questions when used in a lecture format. The audio can be turned off if desired. The movies can be included in assignments prior to classroom discussions or used for self study.

The examples used in the movies are similar to those appearing in many texts for matrix algebra or linear algebra. Thus they involve small matrices with integer entries.

Topic	Time	Comments	Click to view
The Method of Elimination	5 min 40 sec	Step-by-step procedure for a 3 by 3 system.
Matrix Addition	2 min 17 sec	Adds 3 by 2 matrices.
Scalars and Scalar Multiplication	3 min 12 sec	Scalar quantities and multiplication of a matrix by a scalar.
Matrix Differences (subtraction)	2 min 38 sec	Subtraction of matrices.
Linear Combinations	5 min 6 sec	Scalar multiples with addition or subtraction of matrices.
Dot Product	1 min 40 sec	Uses a pair of 4-vectors.
Matrix Multiplication	3 min 13 sec	Uses row-by-column computation.
Matrix-Vector Product	1 min 57 sec	Uses a linear combination of columns format.
Matrix Transformations	7 min 25 sec	Functions defined by matrix-vector products.
Matrix Inverse	15 min 23 sec	Reversing the effect of matrix multiplication, when possible.
Reduced Row Echelon Form (easy)	4 min 20 sec	Step-by-step on an 3 by 4 matrix.
Reduced Row Echelon Form (typical)	7 min 12 sec	Step-by-step on an 4 by 4 matrix.
Solving a 3 by 3 Linear System	4 min 50 sec	Step-by-step solution of 3 equations in 3 unknowns.
Solving a Homogeneous System	4 min 49 sec	Step-by-step on a system of 3 equations in 4 unknowns.
Determinants of Small Matrices (Computing Determinants of 2 by 2 and 3 by 3 Matrices)	5 min 57 sec	Step-by-step development with examples and quizzes.
Determinants by Expansion.	12 min 17 sec	Determinants as a linear combination of determinants of submatrices.
Determinants Using Row Operations	10 min 29 sec	Computing the determinant using row operations.
An Application of Determinants	5 min 42 sec 5 min 58 sec 6 min 1 sec	The Area of a Parallelogram by Determinants and the Method of Elimination	Example 1 Example 2 Example 3
Span Part 1	7 min 49 sec	The span of a set.
Span Part 2	16 min 57 sec	When is a quantity in the span of a set?
Closure	16 min 53 sec	When is a set closed?
Linear Independence/Dependence	17 min 51 sec	Is there a set of coefficients not all zero so that a linear combination of quantities gives zero?
A Discussion about Eigenvalues and Eigenvectors	9 min 43 sec	Geometric and algebraic introduction.
Computing Eigenvalues and Eigenvectors of a 2 by 2 Matrix	6 min 2 sec	Step-by-step computation
Computing Eigenvalues and Eigenvectors of a 3 by 3 Matrix	8 min 7 sec	Step-by-step computation with repeated eigenvalue
Computing Eigenvalues and Eigenvectors of a 3 by 3 Matrix (a second example)	9 min	Step-by-step computation with repeated eigenvalue

If you have MATLAB and want to discuss step-by-step row reduction employing routine reduce, then the following two movies can be incorporated into a presentation, or assigned for students to view independently. In either case we suggest that the students make a list of the row operations used for the reduction so that the process can be reviewed at the end of the movie.

Topic	Time	Comments	Click to view
Solve a 2 by 2 linear system.	3 min 10 sec	Steps are used to avoid the fractions.
Solve a 3 by 3 linear system.	3 min 47 sec	Reduction process uses the addresses of entries in forming fractions for multipliers.

                                                       
 Credits:  This demo was developed by 

David R. Hill
Department of  Mathematics
Temple University

  and is included in Demos with Positive Impact with his permission.

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