Problem Set 8 – Coordinates

Learning Objectives:

• If B is a basis of Rn, you should understand what we mean by the B-coordinates of a vector ~x in Rn, and you should be able to compute [~x]B for a given vector ~x.

• You should be comfortable interpreting the B-matrix of a linear transformation.

• You should understand the relationship between the B-matrix and standard matrix of a linear trans- formation.

• You should be able to use coordinates to find the matrix of a linear transformation, and you should recognize when this is a useful strategy.

The first problem below is a warmup; you need not turn it in.

W1. (a) Verify that the vectors ~v1 =

 11

1

  ,~v2 =

 12

3

  ,~v3 =

 13

6

  form a basis B of R3.

(b) Express the vector ~x =

 22

3

  as a linear combination of ~v1,~v2,~v3.

(c) What is [~x]B?

1. (a) The picture below shows a basis B = (~v1,~v2) of R2, as well as a third vector ~x in R2. What is [~x]B?

~v1

~v2

~x

(b) Suppose we have a basis B = (~v1,~v2,~v3,~v4,~v5) of R5. If ~x = 3~v1 − 2~v4 + ~v5, what is [~x]B?

2. Find a basis B of R2 such that [~e1]B = [ 3 2

] and [~e2]B =

[ −1

1

] .

3. Bretscher #3.4.66

4. Define a linear transformation T : R3 → R3 by letting T (~x) be the reflection of ~x about the plane x1 − 2×2 + 3×3 = 0.

(a) Find a basis B = (~v1,~v2,~v3) of R3 for which you can easily find T (~v1), T (~v2), T (~v3).

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(b) If B is the basis you found in (a), what is the B-matrix B of T ?

(c) Find the standard matrix of T .

(d) 4 Let A be your answer to (c); that is, A is the standard matrix of T . Calculate A~v1, A~v2, and A~v3 to check that you’ve found the matrix A correctly. (Here, ~v1,~v2,~v3 are the basis vectors you chose in (a).)

5. Let T : R3 → R3 be rotation by 180◦ about the line spanned by

 45

6

 . In each part, you are given a

matrix B. Either find a basis B of R3 such that the B-matrix of T is B, or explain why there is no such basis.

(a)

 1 0 00 −1 0

0 0 −1

  (b)

 1 0 00 1 0

0 0 −1

  (c)

 −1 0 00 −1 0

0 0 1

  (d)

 1 0 −10 −1 0

0 0 0

 

Hint: In two parts, there is such a basis; in two, there is not.

6. First, read Definition 9 on the “Coordinates” handout to make sure you understand the definition of similar matrices.

True or false. If the statement is true, explain why; if the statement is false, give a counterexample.

(a) Bretscher Chapter 3 Exercises (pg. 151-152), #4

(b) Bretscher Chapter 3 Exercises (pg. 151-152), #30

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