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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