Linear Algebra I & Mathematics Tutorial 1b Tutorial 6, 10th November, 14:45 - 15:29 menti.com : 13 41 84 Today: Week 5 + Midterm Next week MONDAY(!!) November 16th, 10:30: Midterm exam (here in Zoom) Midterm content: Week 1-6 (do not worry about week 6 so much) This Thursday 12th Nov. 12:00: Additional question session (here in Zoom)

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Linear Algebra I & Mathematics Tutorial 1bNagoya University, G30 Program

Fall 2020

Instructor: Henrik Bachmann

Homework 4: Geometric linear maps

•

Deadline: 22nd November, 2020

Exercise 1. (2+4 = 6 Points) Let u =

0

B@u1...

un

1

CA 2 Rnbe with u 6= 0.

i) Show that the reflection ⇢u : Rn �! Rnis a linear map.

ii) Show that the matrix of the projection Pu : Rn �! Rnis given by

[Pu] =1

u • uuuT 2 Rn⇥n ,

where uT= (u1 u2 . . . un) 2 R1⇥n

. Use this to give an expression for [⇢u].

Exercise 2. (3+3 = 6 Points) Let u =

✓2

1

◆, d =

✓1

1

◆and x =

✓5

0

◆.

i) Calculate the matrices [Pu] and [⇢u] in this special case.

ii) Calculate the following vectors and draw them in one picture together with u,d and x

Pu(x), ⇢u(x), (Pu � Pd)(x), rot⇡2(x), (Pu � rot⇡

2)(x), (rot⇡

2�Pu)(x), (Pd � rot⇡

2�Pu)(x) .

Exercise 3. (3+3 = 6 Points) Show that for all u 2 Rnwith u 6= 0 the projection Pu and the reflection

⇢u satisfy for all x 2 Rnthe following two properties:

i) Pu(Pu(x)) = Pu(x).

ii) ⇢u(⇢u(x)) = x.

Version: November 7, 2020- 1 -

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Linear Algebra I - Midterm examNagoya University, G30 Program

Fall 2019

Instructor: Henrik Bachmann

1) (10 Points) Consider the following linear system

8<

:

2x1 + 3x2 + 4x3 + 5x4 = 6

x1 + 2x2 + 3x3 + 4x4 = 5

3x1 + 4x2 + 5x3 + 6x4 = 7

.

i) Find a matrix A 2 R3⇥4and and a vector b 2 R3

, such that the solutions of the above linear system

are given by the vectors x =

x1x2x3x4

!2 R4

satisfying Ax = b.

ii) Determine the row-reduced echelon forms of the matrices (A | b) and A.

iii) Find all the solutions to the linear system.

iv) Calculate the rank of (A | b) and A.

v) Find a vector c 2 R3, such that Ax = c has no solutions. Calculate the rank of (A | c).

2) (10 Points) Let u =

✓2

1

◆2 R2

and define the following four functions:

f1 : R2 �! R2

x 7�! (u • x)u+ x ,

f2 : R �! R2

x 7�!✓2 cos(x)

sin(x)

◆,

f3 : R3 �! R2

x 7�! x • xu • uu ,

f4 : R3 �! R4

0

@x1

x2

x3

1

A 7�!

0

BB@

0

x1 + 2x2 + 3x3

x1 � x3

x2

1

CCA .

i) Which of the above functions f1, f2, f3, f4 are linear maps? For each one that is linear, determine

its matrix.

ii) Draw a picture of the image of f2. Is f2 injective and/or surjective?

3) (6 Points) Let G : R2 ! R2be a linear map with

G

✓1

1

◆=

✓�1

1

◆, G

✓1

2

◆=

✓3

4

◆.

i) Determine the matrix of G.

ii) Determine the matrix of G �G.

4) (6 Points) We define the following linear map

H : R3 �! R3

0

@x1

x2

x3

1

A 7�!

0

@x1

x2 + x3

x1 + x2 + x3

1

A .

i) Calculate the image of H.

ii) Decide if H is injective and/or surjective.

iii) Find all vectors v 2 R3, which are orthogonal to all vectors in the image of H.

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Mathematics Tutorial 1b (Linear Algebra I)Nagoya University, G30 Program

Fall 2019

Instructor: Henrik Bachmann

Tutorial 6: Review for the midterm exam

•

Exercise 1. Give an example of a linear system which has

i) exactly one solution.

ii) infinitely many solutions.

iii) no solutions.

Exercise 2. Which of the following matrices are on row-reduced echelon form?

0

@1 1

0 1

0 0

1

A ,�1 2 3

�,

0

@1 0 �1 0

0 1 3 0

0 0 0 1

1

A ,

0

@0

0

0

1

A ,

✓1 1 0 1

0 0 1 1

◆.

Exercise 3. Consider the following linear system

8<

:

x1 + 4x2 + 7x3 + 2x4 = 1

2x1 + 5x2 + 8x3 + x4 = 2

3x1 + 6x2 + 10x3 + x4 = 1

.

i) Find a matrix A 2 R3⇥4and and a vector b 2 R3

, such that the solutions of the above linear system

are given by the vectors x =

0

@x1x2x3x4

1

A 2 R4satisfying Ax = b.

ii) Calculate the row-reduced echelon form of (A | b).

iii) Find all the solutions to the linear system.

iv) Calculate the rank of (A | b) and A.

Exercise 4. Define for a matrix A 2 Rm⇥nthe linear map

FA : Rn �! Rm

x 7�! Ax .

Choose for A the matrices appearing in Exercise 2 & 3 and decide for each case if FA is injective and/or

surjective and calculate the image of FA.

Exercise 5.

i) Determine all vectors that are orthogonal to the vector v =

0

@1

2

3

1

A.

ii) Find a linear map G : R2 ! R3, such that the image of G is given by all the vectors which are

orthogonal to v. (i.e. the image gives all the vectors you determined in i) ).

Version: November 14, 2019- 1 -

Exercise 6. Let u =

✓1

2

◆2 R2

and define the following four functions:

f1 : R �! R2

x 7�! rotx(u) ,

f2 : R2 �! R2

x 7�!✓

u • x(u • u)(x • u)

◆,

f3 : R3 �! R2

0

@x1

x2

x3

1

A 7�!✓x1 + x2 + x3

1

◆,

f4 : R2 �! R3

✓x1

x2

◆7�!

0

@2x2 � x1

2x1

4x1 + x2

1

A .

i) Give a geometric interpretation of the image of f1. Is f1 injective and/or surjective?

ii) Which of the above functions f1, f2, f3, f4 are linear maps? For each one that is linear, determine

its matrix.

Exercise 7. Let G : R2 ! R3be a linear map with

G

✓1

�1

◆=

0

@1

2

3

1

A , G

✓0

2

◆=

0

@4

2

2

1

A .

Determine the matrix of G and calculate G

✓1

1

◆.

Exercise 8. Review the exercises of Homework 1, 2, 3 and Quiz 1, 2.

2