In this chapter, students will study various operations using matrices. They will also study how those operations can be applied to geometric transformations and real-life situations.
Matrices first entered high school mathematics programs with the new-math curricula of the late 1950s and early 1960s. Their appearance was justified by their importance to linear algebra in general and to the study of systems of equations in particular. Now, with the increase of importance of computers, two other uses have become apparent—first, in the storage of data, and second, in representing geometric transformations. Thus matrices and their operations should be familiar content.
There are two broad goals for Chapter 4. One goal is to study matrices as a means for storing data and solving problems. A second is to use matrices to review geometric transformations. The language of reflections, rotations, translations, size changes, and scale changes appears again in later chapters. The application to solve systems appears in Chapter 5. This chapter consists of several groups of lessons.
Lessons 4-1 to 4-3 introduce the vocabulary and notation for matrices, their use in storing data, and the operations of matrix addition and multiplication.
Lessons 4-4 and 4-5 discuss matrices used for size and scale changes, along with those properties which are preserved or not preserved.
Lessons 4-6 to 4-8 introduce matrices used for reflections and rotations. These lessons provide the mathematics necessary to prove the relationship between slopes of perpendicular lines presented in Lesson 4-9.
The final lesson deals with matrix addition and its application to the study of translations. A Summary at the end of the chapter contains all the transformation matrices presented in Chapter 4.
The matrix content of Chapter 4 will be new to most students, but most of the corresponding transformations will be familiar to students who have studied Transition Mathematics, UCSMP Algebra or Geometry. Only scale changes will likely be new.
In this chapter, the matrices have numerical entries only. However, many matrices contain alphanumeric data—that is, a combination of letters and numbers.
Objectives
SkillsProperties
- Add, subtract, and find scalar multiples of matrices.
- Multiply matrices.
- Determine equations of lines of perpendiculars to given lines.
Uses
- Recognize properties of matrix operations.
- Recognize relationships between figures and their transformation images.
- Relate transformations to matrices, and vice versa.
Representations
- Use matrices to store data.
- Use matrix addition, matrix multiplication, and scalar multiplication to solve real-world problems.
- Graph figures and their transformation images.
Vocabulary
Lesson 4-1Lesson 4-2
- matrix, matrices
- element of a matrix
- dimensions n
m
- row
- column
- equal matrices
- point matrix
Lesson 4-3
- matrix addition
- sum of matrices
- difference of matrices
- scalar multiplication
Lesson 4-4
- matrix multiplication
- headings
- 2
Lesson 4-5
- standard form
- transformation
- size change, S
- preimage
- image
- similar
- ratio of similitude
- center
- magnitude of size change
- identity transformation
Lesson 4-6
- scale change, Sa,b
- horizontal magnitude
- vertical magnitude
- stretch, shrink
Lesson 4-7
- reflection image of a point over a line
- reflection image
- reflecting line
- line of reflection
- reflection
- rx, ry, rx = y
- Matrix Basis Theorem
Lesson 4-8
- closure
- composite of transformation
- composed
- R90
Lesson 4-10
- rotation
- Rx, R90, R180, R270,
- translation, Th,k
- slide or translation image