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Linear Algebra I & Mathematics Tutorial 1b Tutorial 4, 27th
October, 14:45 - 15:29
You are free to record this Tutorial for your own purposes.
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Next week: Second quiz! (just 2 questions) Topic: Functions,
Image, Injective, Surjective, Bijective (Week 3) (NOT 4)
Alsotoday
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Linear Algebra I & Mathematics Tutorial 1bNagoya University,
G30 Program
Fall 2020
Instructor: Henrik Bachmann
Homework 3: Functions & Linear maps
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Deadline: 8th November, 2020
Exercise 1. (3+4+3=10 Points) We define the following four
functions:
f1 : R �! R2
x 7�!✓cos(x)sin(x)
◆,
f2 : R2 �! R3
✓x1x2
◆7�!
0
@x1 � 2x23x1 + x2x1 � x2
1
A ,
f3 : R �! Rx 7�! 4x� 1 ,
f4 : R2 �! R2✓x1x2
◆7�!
✓x1 + 2x2x1x2
◆.
i) Calculate the image of each function, i.e. describe im(fj)
for j = 1, 2, 3, 4 as explicit as possible.
ii) Decide for each function if it is injective and/or
surjective and/or bijective.
iii) Decide which of the above functions are linear maps.
Justify your answers in ii) and iii).
Exercise 2. (6 Points) Show that there exist a unique linear map
G : R2 ! R3 with the property
G
✓�12
◆=
0
@1
2
3
1
A , G✓
1
�1
◆=
0
@4
5
6
1
A .
What is the value of G(x) for an arbitrary x =
✓x1x2
◆2 R2? Determine the matrix of G.
Exercise 3. (4 Points) Let F : Rn ! Rm be a linear map. Show
that the following two statements areequivalent:
i) F is injective.
ii) The only solution to F (x) = 0 is x =
0
B@
0
.
.
.
0
1
CA.
To show that both statements are equivalent you need to show
that i) implies ii) and ii) implies i).
Version: October 24, 2020
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Today
NextTutorial3rd Nov
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X Y Sets
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