FROM CONCRETE TO ABSTRACT Basic Skills Analysis Hypothesis Proof Elementary Matrices and Geometrical Transformations for Linear Algebra Helena Mirtova Prince George’s Community College

Definitions and Properties Elementary row operations 1. Interchange two rows 2. Multiply a row by nonzero constant 3. Add a multiple of a row to another row An n by n matrix is called an elementary matrix if it can be obtained from the identity matrix I n by a single elementary row operation If E is elementary matrix then E -1 exists and is an elementary matrix Any invertible matrix can be written as the product of elementary matrices

Project Set Up Students work in small groups (2 or 3 people). Each group is given different matrix and geometrical shape. After all groups have completed the basic skills section, they share results with the rest of the class and instructor facilitates further discussion.

1. Write A as a product of elementary matrices

1. Write A as a product of elementary matrices (contd.)

2. Create data matrix D by entering coordinates of the vertices of the given object D1 = D2 =

3. Find the result of a geometrical transformation represented by matrix A A*D1 = A*D2 = Describe what happened to the original shape ________________

4. Break your graphical transformation into series of “elementary” transformations using representation of matrix A as a product of elementary matrixes. Describe each transformation.

Discussion Is there only one way to present a matrix as product of elementary matrices? Can any matrix be factored in elementary matrices? Does order of elementary geometric transformations matter? Which types of elementary transformations did you observe? Can shift along x or y axis be described by any elementary matrix ?

1. List all possible types of elementary matrices 2. Describe basic geometrical transformations defined by each type of elementary matrix and prove your conclusion 1 – reflection about y=x 2 – horizontal expansion or contraction 3 – vertical expansion or contraction 4,5 – horizontal and vertical shear

Sample Proof from Student For any point with coordinates (x,y) Therefore elementary matrix represent reflection about line y=x