Mathematics

Create at least three visuals.One visual must be a scatter plot with trend line, equation, R2 value, and prediction value.Two of the remaining required visuals can be of the following format: histogram, box and whisker plot, or pie chart.Please note that the data set that you chose in Week 2 includes data that will not be needed to create your visuals. Quantitative reasoning requires critical thinking to decide what data is necessary.  Create a Microsoft® Word document that includes your three visuals and the following items:Title of your project and the scenario you are addressingBrief description of each visual (15 to 50 words)Consider including the following for each visual when applicable:A chart title that is appropriate for the dataA descriptive x-axis labelA descriptive y-axis labelFor your xy scatter plot, make at least one prediction using the trend line equation for a date in the future. How confident are you in this prediction? State your prediction and provide justification (50 to 150 words). If you created a box and whisker plot, describe the central tendency of the values. What does this tell you about the data and about your project?Calculate the mean of the sample data.

Problem Set 5 – More Examples of Linear Transformations

Learning Objectives:

• You should be familiar with several geometric examples of linear transformations (rotations, scalings, reflections, projections, and shears): you should be able to visualize these linear transformations and find their matrices. However, don’t forget that most linear transformations don’t have a simple geometric interpretation!

• You should understand the very important principle that a linear transformation is determined by what it does to a basis of its domain. In particular, you should see how we use this principle to find the matrices of the geometric linear transformations.

1. Bretscher #2.2.8

2. (a) Bretscher #2.2.20

(b) Bretscher #2.2.22

In case it is helpful, here is a picture showing the positive x-, y-, and z-axes.

x

y

z

3. Let L be the line spanned by

[ 2 5

] . In this problem, you’ll look at the linear transformation projL.

(a) Find a basis (~v1, ~v2) of R2 for which you can very easily find projL(~v1) and projL(~v2).

(b) Use the basis you chose in (a) to help you find the matrix of projL. (If you’re stuck, take another look at Problem Set 3, #2.)

4. Eventually, we’ll want to consider orthogonal projections in Rn for n > 2 (yes, even in spaces like R100!). This problem is meant to help you think about orthogonality from an algebraic perspective.

In this problem, you’ll find all vectors in R3 perpendicular to both

12 3

 and  15 −9

. If you took Math 21a, you’re probably used to using the cross product in problems like this. However, that approach works only in R3, so we’ll have no way to generalize it to R4 or R17. Here’s a different approach that can be generalized to Rn for any n.

(a) If

xy z

 is perpendicular to 12

3

, what is x + 2y + 3z equal to? (Hint: x + 2y + 3z is exactly the 1

dot product

xy z

 · 12

3

.)

(b) If

xy z

 is perpendicular to  15 −9

, what is x + 5y − 9z equal to?

(c) You now have a system x + 2y + 3z = ? x + 5y − 9z = ? which exactly expresses the fact that

xy z

 is perpendicular to both

12 3

 and  15 −9

. Solve this system to find all vectors perpendicular to both

12 3

 and  15 −9

. 5. Next time in class, we will be talking about the composition of linear transformations. Be prepared to

discuss this problem in class.

(a) Bretscher #2.1.45 (Let us know whether the transformation preserves addition and scalar multi- plication.)

(b) Bretscher #2.1.46

2

 

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