Thursday HW Notes



4-1 Homework Notes

4-1: Sorting data in Matrices

Information is often stored in __________________. The inventory of athletic clothing owned by a high school cross-country team is shown in the matrix below:

Pants Shirts Shorts

Small row ___

Medium row ___

Large row ___

x-large row ___

the element in the _______ row and the ______ column

This matrix has _____ rows and ______ columns. It is said to have the _________________________ 4 by 3, written _____________. In general, a matrix with ____ rows and ___ columns had dimensions __________.

Example: The Matterhorn Company produced 1500 trumpets and 1200 French Horns in September; 2000 trumpets and 1400 French Horns in October; and 900 trumpets and 700 French horns in November.

a) Store the company’s products in a matrix. b) What are it’s dimensions?

4-1 In-Class Notes

Points and polygons can also be represented by _____________________. The ordered pair (x, y) is generally represented by the matrix [pic]. This is called a __________________________________.

Read through p. 8 in the pink Core Plus books that are on your tables. Complete questions 5-7 with your neighbors in the space below:

If time, read through Investigation 3 on p. 10 and answer questions 1 & 2 together.

HW: Complete p. 207 #2,4-7,9,10 on a separate sheet

4-2 Homework Notes

4-2: Matrix Addition

There are many situations which require ______________ the information stored in matrices. For instance, suppose matrix C represents the current inventory of the Chic Boutique.

8 10 12 14 16

dresses

suits

skirts

blouses

The quantities of new items received by the boutique are represented by the numbers stores in matrix D.

8 10 12 14 16

dresses

suits

skirts

blouses

The new inventory is found by taking the _____ of matrices ____________. This ______________________ is performed according to the following definition:

If two matrices ____ and _____ have the same ___________________________, their ______ A + B is the matrix in which each element is the ______ of the corresponding matrix in which each element is the sum of the corresponding elements in A and B.

Complete the matrix for the new inventory

8 10 12 14 16

dresses

suits

skirts

blouses

Addition of matrices is __________________, meaning that A + B = ___________. It is also _______________________ meaning (A + B) + C = ___________________. Subtraction works the same way.

Scalar Multiplication

The __________________ of a scalar k and a matrix A is the matrix _____ in which each element is ____ times the corresponding element in A.

Example: Find the product

Complete p. 211 #1,2

4-2 In-Class Notes

Read through p. 11-12 (under the car picture) in the pink Core Plus books. Complete questions #3-6 with your neighbors in the space below:

Read through p. 13 (On Your Own) in the pink Core Plus books. Complete the questions with your neighbors in the space below:

From p. 20 #3 “A square matrix is said to be _______________________ if it has symmetry about its ____________ _______________________. The main diagonal of a square matrix is the ____________________ line of entries running from the ________ ________ to the ____________________ _____________ corner.” Answer question #3c together as a class.

Monday Homework: p. 211 #5,6,7,10 on a separate sheet.

4-3 Homework Notes

4-3: Matrix Multiplication

Matrix multiplication is much more complex than Matrix addition. You can not just multiply the individual elements. Before you try to do the multiplication, it would be best to write down the definition first.

Definition of Matrix Multiplication

Suppose A is an ___________ matrix and B is an ____________ matrix. Then the product ______________________ is the __________ matrix whose element in row ____ and column _____ is the product of _________________ and ___________________.

This means that the product of two matrices __________ exists only when the number of ___________________ of A equals the number of ____________ of B. It also means that the dimensions of the answer matrix will have the same number of ___________________ of A equals the number of ____________ of B.

Example: Multiply the following:

First write down the dimensions of the first two matrices to make sure that the product exists and to find the dimensions of the answer matrix.

To get the solution for the element in the 1st row and the 1st column, you multiply across the _____________ of Matrix A and down the ____________________ of Matrix B. You then take the product of the 1st elements and the product of the 2nd elements and add them together.

Fill this into the answer matrix above. Then complete this process to fill in the rest. Show your work below.

Example: Can you multiply

Why or why not?

This indicates that, in general, multiplication of matrices is not ________________________________.

HW: Complete p. 217, #1-3

4-3 In Class Notes

When given a story problem, make sure that you set up your matrices so that they will be able to be multiplied.

Example: Costumes have been designed for the school play. Each boy’s costume requires 5 yards of fabric, 4 yards of ribbon, and 3 packets of sequins. Each girl’s costume requires 6 yards of fabric, 5 yards of ribbon, and 2 packets of sequins. Fabric costs $4 per yard, ribbon costs $2 per yard, and sequins cost $.50 per packet. Use matrix multiplications to find the total cost of the materials for each costume.

Now I will show you how to do this on your calculator. Take notes below if needed.

Read through p. 30 (On Your Own) in the pink Core Plus books. Complete the questions with your neighbors in the space below:

HW: Complete p. 217, #4-7, 10-12 on a separate sheet as well as WS Master 10

Geometry Transformation In-Class Notes

The Alhambra, a 13th-century palace in Grenada, Spain, is famous for the geometric patterns that cover its walls and floors. To create a variety of designs, the builders based the patterns on several different transformations.

A ___________________ is a change in the position, size, or shape of a figure. The original figure is called the _____________. The resulting figure is called the __________ A transformation maps the preimage to the image. Arrow notation (() is used to describe a transformation, and primes (’) are used to label the image.

Example 2: Drawing and Identifying Transformations

A figure has vertices at A(1, –1), B(2, 3), and C(4, –2). After a transformation, the image of the figure has vertices at A'(–1, –1), B'(–2, 3), and C'(–4, –2). Draw the preimage and image. Then identify the transformation.

Check It Out! Example 2

A figure has vertices at E(2, 0), F(2, -1), G(5, -1), and H(5, 0). After a transformation, the image of the figure has vertices at E’(0, 2), F’(1, 2), G’(1, 5), and H’(0, 5). Draw the preimage and image. Then identify the transformation.

To find coordinates for the image of a figure in a translation, add a to the x-coordinates of the preimage and add b to the y-coordinates of the preimage. Translations can also be described by a rule such as (x, y) ( _______________.

Example 3: Translations in the Coordinate Plane

Find the coordinates for the image of ∆ABC after the translation (x, y) ( (x + 2, y - 1). Draw the image.

Example 4: Art History Application

The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2.

[pic]

A ___________ is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A __________ ____________ describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) ( (ka, kb).

[pic]

Example 1: Computer Graphics Application

Draw the border of the photo after a dilation with scale factor 5/2 .

Check It Out! Example 1

What if…? Draw the border of the original photo after a dilation with scale factor ½.

[pic]

Complete Worksheet

4-4 & 4-5 Homework Notes

4-4: Matrices for Size Change

A __________________________ is a one-to-one correspondence between the points of a ___________________ and the points of an _______________. Consider ∆PQR with P = (3, 1), Q = (-4, 0), and R = (-3, -2). The size change with center ________ and magnitude 3, denoted ________, can be performed by ________________________ each x- and y- coordinate on ∆PQR by ____. We write:

Recall that the symbol ________ is often used in mathematics to denote “_______________”.

For any k ≠ 0, the transformation that maps ________ onto ___________ is called the ________________________ with center _________ and _______________________ k, and is denoted ________.

Size change images can also be found by multiplying _______________. When the matrix for the point __________ is

multiplied by ______________, the matrix for the point ___________ results:

Theorem: ___________ is the matrix for _______.

Example 1: Given ABCD with A = (0, 3), B = (-2, -4), C = (-6, -4), and D = (-6, 4), find the image of A’B’C’D’ under S4.

4-5: Matrices for Scale Change

For any nonzero numbers a and b, the transformation that maps ___________ onto ___________ is called the ________________________ with ___________________________________________ a and _____________________ ____________________________ b, and is denoted _______.

Theorem: is a matrix for ________.

Complete the work on p. 228 under “Matrices for Scale Change” to prove this.

Example: Refer to the quadrilateral from the previous example. Use matrix multiplication to find its image under S2,5.

4-4 & 4-5 In-Class Notes

4-4: Matrices for Size Change

The transformation that maps each point __________ onto itself is called the __________________________________.

When a point matrix is multiplied on the _________ by each point __________ coincides with the image. In other words, multiplying by the identity matrix is the same as performing a size change of magnitude _____.

Example: Refer to the quadrilaterals in the 1st example from the homework notes. Calculate each ratio.

[pic] [pic]

Find the mapping equations for both a size change as well as a scale change.

Homework: p.224 #4, 7-14, 18 and p.229 #1,9,11 on a separate sheet of paper.

4-6 Homework Notes

4-6: Matrices for Reflections

A _____________________ is a transformation that maps a figure to its reflection image. Reflection over the y-axis can be denoted as ____________ or ______. The mapping equation for this is ___________________. Reflection over the x-axis can be denoted as _____________ or ______. The mapping equation for this is ____________________. Reflection over the line y = x can be denoted as _____________ . The mapping equation for this is ________________________.

List the matrices for these three below:

Example: If A = (1, 2), B = (1, 4), and C = (2, 4), find the image of ∆ABC under [pic], [pic], and [pic]. Show all 4 on the same graph.

4-6 In-Class Notes

4-6: Matrices for Reflections

Remembering Matrices

At this point, you have seen matrices for some size changes, some scale changes, and three reflections. You may wonder, is there a trick to remembering them? Here is one great way:

Pretend that you have two points, (1, 0) and (0, 1), giving you the matrix [pic] (the identity matrix). If you can figure out where these two points will go when you complete the given transformation, you will get the matrix you need.

Example: Complete this for all of the transformations you have done so far.

4-8 Homework Notes

4-8: Matrices for Rotations

_______________ are closely related to __________. The arcs used to denote angles suggest turns. Angles with larger measures indicate greater ___________. The amount and direction of the turn determine the magnitude of the _____________. The rotation of magnitude ___ around the ____________ is denoted ____.

Caution: A rotation is denoted with a ______________________, while a reflection is denoted with a ______________________________.

Rotations often occur one after the other. A _______________ of ______ following a rotation of ____ with the same center results in a rotation of ____________. In symbols, ________________.

Use the trick you learned in class to figure out the matrices for 90˚, 180˚, 270˚, and 360˚.

4-8 In-Class Notes

4-8: Matrices for Rotations

Prove that you can get [pic]by multiplying [pic]and [pic].

Prove that you can get [pic]by multiplying [pic]and [pic].

Prove that you can get [pic]by multiplying [pic]and [pic].

Homework: p. 248 #1-6, 11-14 on a separate sheet

4-10 Homework Notes

4-10: Matrices for Translations

The transformation that maps ___________ onto ____________________________ is a _________________________ of ____ units ______________________ and ____ units ____________________________ and is denoted ________.

Since we have to add numbers to x and y, we will have to add matrices instead of multiply. This means that the dimensions of the transformation matrix must be ____________________ as the point matrix.

Example: A quadrilateral has vertices Q = (-4, 2), U = (-2, 6), A = (0, 5) and D = (0, 3). Put these points into a matrix.

What must the dimensions of the translation matrix be? ___________ Write the matrix for [pic] below.

Add [pic] to QUAD to find the matrix for Q’U’A’D’. Then graph both the preimage and the image.

5-5 Homework Notes

5-5: Inverse of Matrices

Review:

Plot quadrilateral ABCD with A = (-2, 1), B = (1, 1), C = (1, -2), and D = (-2, -2). It will help to put the points in a matrix first.

Next find the matrix for the image of that shape under [pic]. Then plot this on the same graph.

Now find the matrix for the image of the shape from #2 under [pic]. Plot this. How does this compare to the original shape?

Multiply the matrices from #2 and #3. Go back to your Chapter 4 Notes if you forgot how to multiply matrices. What is this called?

Two matrices whose product is the _____________________________ are called __________________________.

5-5 In-Class Notes

In general, only ________________ matrices can have inverses, and two matrices are only inverses if and only if their product is the ___________________ matrix.

Example: Verify that the inverse of [pic]is [pic]

Write down some notes from the inverses from the previous examples that may help you figure out how to find the inverse of a matrix.

Unfortunately, it is not always as simple as the examples above. Below is the Inverse Matrix Theorem:

If [pic], the inverse of [pic] is

Example: Find the inverse of the matrix [pic]

Not all square matrices have inverses. If [pic]does in fact equal _____, then all of the numbers in the inverse will be _____________________. Since [pic]is so important, it has a special name, the _________________________.

Example: Determine whether [pic] has an inverse or not.

HW: p. 3.2-3.3, #1-3, 5-9, 16 on a separate sheet

-----------------------

[pic]

Column ____

Column ____

Column ____

[pic]

[pic]

[pic]

[pic]

[pic]

Column 1

[pic]

Row 1

Row 1

[pic]

Column 1

[pic]

[pic]

A

A’

If the scale factor of a dilation is greater than 1 (k > 1), it is an _______________. If the scale factor is less than 1 (k < 1), it is a _________.

Helpful Hint

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