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FROM CONCRETE TO ABSTRACTACTIVITIES FOR LINEAR ALGEBRABasic Skills Analysis Hypothesis Proof
Helena Mirtova
Prince George’s Community College
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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
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Welcome to Spy Alley!
Please, distribute cards with your code names and communication protocols.
You have 10 minutes to get to checkpoint 1
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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
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Mata Hari
Mrs. Peele
Mr. Steele Severus Snape
Spy Dude
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2. Create adjacency matrix A for your digraph, where each element is defined by
1 if there is arc from vertex to vertex
0 if there is no connecting arc
0 if ij
i j
a
i j
3. Find A2, A3, and A4
Checkpoint 1
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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 Am )(m
ija
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 Vi to Vj describe it!
)(mija
1 - path from V5 to V3
2 – path from V5 to V3
3 – path from V5 to V3
4 – path from V5 to V3
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Discussion of a proof)(k
ija)(k
ijaTheorem: If A is adjacency matrix of a graph and represents (i,j)
entry of Ak, then is equal to the number of walks (paths) of
length k from Vi to Vj
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 Vi to Vj
If statement is true for k =m, show that it is true for k =m+1
Let us first consider the following example:
ija
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What do we see?
The number of 3-paths from V5 to V1 multiplied by 0 because there is no arch from V1 to V3 = 0*0
Added to
The number of 3-paths from V5 to V2 multiplied by 0 because there is no arch from V2 to V3 =4*0
Added to
The number of all 3-paths from V5 to V3 multiplied by 0 because there is no arch from V3 to V3 = 0*0
Added to
The number of all 3-paths from V5 to V4 multiplied by 1 because there one arch from V4 to V3 =2*1
Added to
The number of all 3-paths from V5 to V5 multiplied by 0 because there is no arch from V5 to V3 =1*0
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Generalizing the observation we can interpret (m+1)-path as m-path followed by arch.
Therefore number of (m+1)-paths between Vi and Vj 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 Vi to Vj
Next time we will talk about coding messages in our spy network!
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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.
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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.
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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
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Checkpoint 1
1.
2.
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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 In 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
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3. Write A as a product of elementary matrices
3 1
1 2A
1 2R R
1 2
3 1
1
0 1
1 0E
1
1
0 1
1 0E
2 2 13R R R 2 2 13R R R
1 2
0 5
2
1 0
3 1E
1
2
1 0
3 1E
1 2R R
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2 21
5R R
2 25R R
1 2
0 1
3
1 0
10
5
E
1
3
1 0
0 5E
1 1 22R R R 1 1 22R R R
1 0
0 1
4
1 2
0 1E
1
4
1 2
0 1E
4 3 2 1E E E E A I 1 1 1 11 2 3 4A E E E E
3 1 0 1 1 0 1 0 1 2
1 2 1 0 3 1 0 5 0 1
3. Write A as a product of elementary matrices (contd.)
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4. Break your graphical transformation into series of “elementary” transformations using representation of matrix A as a product of elementary matrixes.
Describe each transformation.
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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 2
0 11 R R
1 0
1 1
2 2
02 R = kR
0 1
1 03 R = kR
0
k
k
1 1 2
2 2 1
14 R = R +kR
0 1
1 05 R = R +kR
0
k
k
1 – reflection about y=x2 – horizontal expansion or contraction
3 – vertical expansion or contraction4,5 – horizontal and vertical shear
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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 ?
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Sample Proof from StudentFor any point with coordinates (x,y)
0 1 =
1 0
x y
y x
Therefore elementary matrix represent reflection about line y=x