Linear Algebra I & Mathematics Tutorial 1b
Nagoya University, G30 Program, Fall 2023

Instructor: Henrik Bachmann
Teaching assistant: Yuichiro Toma

Homework 3: Functions & Linear maps
•Deadline: 12th November, 2023
Exercise 1. (3+3+4=10 Points) We deﬁne the following four functions:f1 : R −→ R2()
1 − cos(x)x 7−→,sin(x)f2 : R2 −→ R2( )()x1
2x1 − x2
7−→,x2x1 x2
f4 : R2 −→ R3
( )3x1 − x2x17−→ 2x1 − x2  .
x2
x1 − x2

f3 : R −→ Rx 7−→

4xx2 + 4,(i) Calculate the image of each function, i.e. describe im(fj ) for j = 1, 2, 3, 4 as explicit as possible. If
you can not ﬁnd a mathematical description try to describe the elements of the image in words.
(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.

Exercise 2. (5 Points) Show that there exist a unique linear map T : R2 → R3 with the property  ( )( )
1
411


2T=,T= 5 .
1−136(What is the value of T (x) for an arbitrary x =x1x2)∈ R2 ? Determine the matrix of T .

Exercise 3. (2+2+3=7 Points)(i) Let X be a ﬁnite set. Show that a function f : X → X is injective if and only if it is surjective.
(ii) Let F : R2 → R3 be a linear map. Show that F can not be surjective.
(iii) Let F : Rn → Rm be a linear map. Show that F is injective if and only if the only solution toF (x) = 0 is x = 0.

Version: October 26, 2023

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Hello ∼ クマ先生 here. With another homework comes another Japanese lesson∼
This week, I’ve prepared ﬁve words: two nouns and three adjectives. First, the nouns:
しゅうごう

しゃぞう

集合写像
せんけいしゃぞう

The two words are shuugou (set) and shazou (map, as in linear map (線形写像)). This time, bothwords are used almost exclusively used in mathematics. Next, we have three words to describe a function:
たんしゃ

ぜんしゃ

ぜんたんしゃ

単射

全射

全単射

These three words are: tansha (injective), zensha (surjective), and zentansha (bijective). To describe
たんしゃしゃぞう

a function (in Japanese), we combine the words. For example, injectiive function is 単射写像.
かんじAnyway, now, a breakdown of the individual 漢字 that makes up these words:
しゅう集-あ----

しゃしん

This kanji means ”image”. Alternatively, it also means statue, picture orぶつぞう

likeness (of ). One word that uses this kanji is 仏像 (Buddha Statue)
This kanji means ”(to) shoot” or ”(to) project”. This refers to how functions”shoot” elements of one set to elements of another set. This kanji is not common
in everyday life.

たんごin the domain. Everyday words that use this kanji include 単語 (vocabulary),
かんたんたんいたんか簡単 (simple), 単位 (credits), and 単価 (unit price).
This kanji means ”all”. It refers to how in a surjective function, every elementin the codomain are associated with (”shot by”) at least one element in theぜん全This kanji means ”(to) copy”. Perhaps, it is used in the word for ”map” since
it ”copies” one set onto another (by associating one element in the domain toThis kanji means ”single. It refers to how in an injective function, everyelement in the range are ”shot by” (or associated with) exactly one element

たん単ごう

one in the codomain). A common use of this kanji is 写真 (photo).

しゃ射are assumed to be similar). Everyday words that include this kanji include 間に合う (to make (it) in time) and 合コン (Matchmaking Party).

ぞう

像

しゅうちゅう

ま-

しゃ

写

あつ

Common uses of this kanji is 集まる (to gather / collect) and 集 中 (focus).
This kanji means ”(to) match”. In a sense, sets are comprised of things (thatごう合This kanji means ”(to) gather”. It refers to how sets are gatherings of things.

まった-すべdomain. This kanji has multiple readings ( 全 く and 全て), and almost all ofthem have the same meaning (”every”). Common words that use this kanji
かんぜんあんぜん

ぜんぜんinclude 完全 (perfect), 安全 (safe), and 全然 (not at all).
And that’s it for today’s (Mathematical) Japanese word(s). またね ∼2