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

Instructor: Henrik Bachmann
Teaching assistant: Yuichiro Toma

Homework 1: Linear systems•Deadline: 15th October, 23:55, 2023
Exercise 1. (2 Points) Try to solve the exercises below and write the solutions down by hand (paper,tablet) or by computer (Latex only. No word!). Write your name, the homework number and the course
name on the ﬁrst page of your solution. Create one pdf-ﬁle (for example, by using a scanner app on
your phone) and submit it before the deadline ends in TACT at the Assignment ”Homework 1”. Use
exactly the following format as a ﬁlename: ”Familyname Givenname LA1 HW1.pdf”. Do this forevery coming Homework to not lose any points.
A linear system is said to be on row-reduced echelon form if the following three conditions are
satisﬁed:(i) The ﬁrst (that is, the leftmost) variable in each equation has coeﬃcient 1.
(ii) If xi is the ﬁrst variable in one of the equations, then it does not occur in any other equation in thesystem.
(iii) If xi is the ﬁrst variable in one equation, then the equations below it do not contain any of thevariables x1 , x2 , . . . , xi−1 .

Exercise 2. (2+2+2+2+2 = 10 Points) Which of the following linear systems are on row-reduced
echelon form? For those that are not, ﬁnd an equivalent system (i.e. one which has the same solutions)that is on row-reduced echelon form. For each system, ﬁnd all solutions.
{x1 + x2 + x3 + 2x4 = 0i)x2−x4 = 0
7x3 = 1
 x1 + 4x2 +8x3 = 2
2x1 + 5x2 +ii)
3x1 + 6x2 + 10x3 = 1{x1 + 2x2 + 3x3 + 4x4 = 2023iii)iv)v)

 x1 = 2x2 = 0x3 = 2{x1 + 3x2
3x1 + 9x2==1
2

Exercise 3. (6 Points) Decide for which real numbers a ∈ R the following linear system has solutions.
Give all the solutions in these cases.
 2x1 + 12x2 + 7x3 = 12a + 7
2x1 +4x2 + 2x3 =12a .
x1 + 10x2 + 6x3 = 7a + 8

Exercise 4. (0 Points) Learn the Kanjis related to this course on the back of each Homework.
(These study sheets were created by Vic Austen).

Version: October 5, 2023

1

せんせい

Welcome to Nagoya University’s Linear Algebra I Course ! My name is クマ先生, and I’m here to teachyou some mathematical Japanese words that will probably (not) be helpful to you guys while living here.
Now let’s get down to business. Today’s word is..
せんけいだいすうがく

線形代数学
senkei daisuugaku
This word literally means linear algebra. Now you know how to say this course’s name in Japanese∼
And then you can say this sentence in your next Japanese class:わたし

せんけいだいすうがく

だい す

私 は線形代数学が大好きです。
かんじ

Anyway, now, a breakdown of the individual 漢字 (kanji) that makes up this word:せん線-This kanji means ”line”. It refers to the fact that this branch of algebra isinterested with ”linear” functions. In everyday life, this kanji can be found in
めいじょうせん

ひがしやません

subway line names (examples in Nagoya: 名 城 線 and 東 山 線)
This kanji means ”form”. If written after another word, the overall word has
せんけいけい形-the meaning ”similar to (word before it)”. In this case, 線形 means ”similarにんぎょう

to line”, or ”linear”. A common (everyday) example is 人 形 (ningyō), whichmeans ”doll”, which are traditionally made in the likeness of humans.
だいすうがくだい代-

This kanji means ”change”. In the sense that it is used, it refers to how 代数学(algebra) ”changes” numbers to letters and/or other symbols. A common use
じ だいof this kanji is in 時代 (jidai), which means era, a ”change” of the times.
すうがく

すう数-

すうじ

fact, the word ”digit” in Japanese is 数字, literally meaning ”number word”.
This kanji means ”study”. This kanji is very common, used in a large number

がく学This kanji means ”number”. In a sense, 数学 (mathematics) is a study of”numbers”. This character usually appears when numbers are involved. Inがっこう-だいがく

かがく

がくせい

of everyday words, such as 学校 (school), 大学 (university), 科学 (science), 学生
りゅうがく(student), and 留 学 (studying abroad).

Now we see that the Japanese word for linear algebra is an apt description of the ﬁeld of study.
And that’s it for today’s (Mathematical) Japanese word. またね ∼2