Week 2 Homework

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

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