1. Why is the state bombarding the curriculum with so many matrices??
Katrina Palmer
Appalachian State University
October 11, 2007

2. Courses with Matrices Algebra I Geometry Algebra II Tech Math I and II
Discrete Math
Integrated Math 2 and 3

3. Outline A few, brief, motivating real world examples
Algebra I Examples
Discrete Math Example
Imaging Example

4. Brief Real World Examples
Facial recognition
Movies
Medical imaging

5. Facial Recognition
What is it?
Why important?

7. (10/8/07)

8. Movies - Yoda Demo

31. "In computer graphics the first thing you want to learn is linear algebra," Sung said. "It's the most important thing." But matrices and vectors were scaring off prospective pupils.
Kelvin Sung, an associate professor of computing and software systems

32. Medical Imaging
CT, MRI, PET
Why important?

33. CT details What is sent through the body?
What kind of data is collected?
A detector detects
how much energy is
left in the ray
X-ray contains
a certain amount
of energy
So we know how much was absorbed

34. Mathematical Relationship (Beer’s Law)Ni No m x

35. Exponential Attenuation of X-rayNi No   

36. Reconstruction Idea 6 4  2 7 3 4 3

37. One Solution
=4
2=3
3=2
4=1

38. Algebra I
Competency Goal 3: The learner will collect, organize, and interpret data with matrices and linear models to solve problems.
3.01 Use matrices to display and interpret data.
3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems.

39. Example from Algebra I
The Watauga volleyball team has 6 wins and 9 losses, Alexander Central has 14 wins and 2 losses, and Avery has 12 wins and 5 losses.

41. Related Questions Write each table as a matrix
Write a matrix that represents the change in participation from 1987 to 1992
Which categories gained the most number of participants? How many?
If you owned a sports store, which sports items would you stock up the most? Why?

42. Discrete Math
Competency Goal 1: The learner will use matrices and graphs to model relationships and solve problems.
1.01 Use matrices to model and solve problems.
Display and interpret data.
Write and evaluate matrix expressions to solve problems.
1.02 Use graph theory to model relationships and solve problems.

43. Google
What is it?
What makes Google™ special?
Why important?

44. Linear Algebra of Google
(hyperlink matrix)

45. Google stochastic matrix (largest eigenvalue = 1)
Eigenvector that corresponds to the eigenvalue of 1 provides that ranking of the importance of each page.

46. Recursive way… Step 1: pick a starting vector, such as [1,1,1,1].
Step 2: Multiply the hyperlink matrix by your starting vector,
Step 3: Multiply the hyperlink matrix by your new vector from step 2.

47. Step 4: Repeat step 3 until the vector repeats itself (becomes stable).

48. What else could a matrix represent?

49. Black and White Images
entry values:
0 = black
255 = white

50. Some Questions How could you darken or lighten an image?
What happens if you zoom in on an image?

51. Zooming

54. Conclusion
“linear algebra is like oatmeal, not much to it, but it will never let you down, and is frequently useful.”

55. Algebra II
Competency Goal 1: The learner will perform operations with complex numbers, matrices, and polynomials.
1.04 Operate with matrices to model and solve problems.
Competency Goal 2: The learner will use relations and functions to solve problems.
2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

56. Geometry
Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.
3.02 Use matrix operations (addition, subtraction, multiplication, scalar multiplication) to describe the transformation of polygons in the coordinate plane.

57. Tech Math I
Competency Goal 3: The learner will describe the transformation of polygons in the coordinate plane algebraically.
3.03 Use matrix operations (addition, subtraction, multiplication, scalar multiplication) to describe the transformation of polygons in the coordinate plane.

58. Tech Math II
Competency Goal 2: The learner will use relations and functions to solve problems.
2.04 Use systems of equations and inequalities to model and solve problems using tables, graphs, matrix operations, and algebraic properties; justify results.

59. Integrated Math 2
Competency Goal 1: The learner will perform operations with numbers, algebraic expressions, and matrices to solve problems.
1.02 Use algebraic expressions, including iterative and recursive forms, to model and solve problems.
1.04 Operate with matrices to model and solve problems.

60. Integrated Math 3
Competency Goal 3: The learner will use relations and functions to solve problems.
3.01 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.