FROM CONCRETE TO ABSTRACT ACTIVITIES FOR LINEAR ALGEBRA Basic Skills Analysis Hypothesis Proof Helena Mirtova Prince George’s Community College

Project 1: Spy communication network Goal: Introducing students to other uses of matrices and matrix operations than in solving systems of equations, enhancing the idea of proof and introducing proof by mathematical induction. Timeline: Project given in the second week of class after solving systems by Gaussian -Jordanian method and right after the formal introduction of matrices and basic matrix operations. Length: 20 minutes of group work & 10 minutes of proof discussion

Welcome to Spy Alley! Please, distribute cards with your code names and communication protocols. You have 10 minutes to get to checkpoint 1

If two agents have the same recognition sign then the agent with the larger sign can send a message to the agent with a small sign via dead letter drop. Different signs refer to different dead letter box locations. A dead drop or dead letter box is a method of espionage tradecraft used to pass items or information between two individuals (e.g., a case officer and agent, or two agents) using a secret location, thus not requiring them to meet directly and thereby maintaining operational security. 1.Draw the digraph (directional graph) of your communication network A digraph is a finite collection of vertices (agents) together with directed arcs joining certain vertices. A path between vertices is a sequence of arcs that allows one to pass messages from one vertex (agent) to another. The length of path is its number of arcs. A path of length n is called n-path

Mata Hari Mrs. Peele Mr. Steele Severus Snape Spy Dude

2. Create adjacency matrix A for your digraph, where each element is defined by 3. Find A 2, A 3, and A 4 Checkpoint 1

4. Explore the graph and the matrix ( you have 10 minutes to get to checkpoint 2): Let be element in row i and column j of A m a) What are the dimension ’ s of matrix A ? b) Do you observe any symmetry in matrix A ? c) Find all: d) If you see any connection between and m – paths from V i to V j describe it! 1 - path from V 5 to V 3 2 – path from V 5 to V 3 3 – path from V 5 to V 3 4 – path from V 5 to V 3

Discussion of a proof Theorem: If A is adjacency matrix of a graph and represents (i,j) entry of A k, then i s equal to the number of walks (paths) of length k from V i to V j Let us use mathematical induction Case k =1, it follows from the definition of the adjacency matrix that represents the number of walks of length 1 from V i to V j If statement is true for k =m, show that it is true for k =m+1 Let us first consider the following example:

What do we see? The number of 3-paths from V 5 to V 1 multiplied by 0 because there is no arch from V 1 to V 3 = 0*0 Added to The number of 3-paths from V 5 to V 2 multiplied by 0 because there is no arch from V 2 to V 3 =4*0 Added to The number of all 3-paths from V 5 to V 3 multiplied by 0 because there is no arch from V 3 to V 3 = 0*0 Added to The number of all 3-paths from V 5 to V 4 multiplied by 1 because there one arch from V 4 to V 3 =2*1 Added to The number of all 3-paths from V 5 to V 5 multiplied by 0 because there is no arch from V 5 to V 3 =1*0

Generalizing the observation we can interpret (m+1)-path as m-path followed by arch. Therefore number of (m+1)-paths between V i and V j is equal to the sum m-paths through all intermediate vertices multiplied by 0 if it is a dead-end or 1 if the (m+1) path can be finished Therefore, represents the total number of (m+1)-paths from V i to V j Next time we will talk about coding messages in our spy network!

Project 2: Elementary Matrices and Geometrical Transformations Goal: Find the matrix of a 2D graphic transformation and present it as a product of elementary matrices. Explore possible types of elementary matrices and basic elementary transformations defined by them. Timeline: Project given close to the end of the course in a linear transformations section. Length: 20 minutes of group work & 10 minutes of proof discussion.

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

1. Create data matrices D1 and D2 by entering coordinates of the vertices of Shape 1 and Shape 2 Shape 1 Shape 2 The data matrices should have the following structure 2. Is shape 2 a linear graphical transformation of shape 1? If yes, find the matrix of this transformation You have 5 minutes to get to checkpoint 1

Checkpoint

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

3. Write A as a product of elementary matrices

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

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

5. List all possible types of elementary matrices 6. 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

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 ?

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