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

Instructor: Henrik Bachmann
Teaching assistant: Yuichiro Toma

Homework 2: Matrices & Vectors•Deadline: 29th October, 23:55, 2023

Exercise 1. (2+2 = 4 Points) Show that for all A ∈ Rm×n , x, y ∈ Rn and λ ∈ R we have(i) A(x + y) = Ax + Ay,(ii) A(λx) = λ(Ax).
(Without using Proposition 2.4. from the lecture).

Let A ∈ Rm×n be a matrix. The rank rk(A) of A is the number of pivot elements in rref(A).
For example,    
11 1 0
1 1 0
1 1 0
1 1 0
1 0
2
A = 2 0 1 ∼ 0 −2 1 ∼ 0 −2 1 ∼ 0 1 − 12  ∼ 0 1 − 12  = rref(A)1 −1 10 −2 1
0 0 0
0 0 00 0 0
and therefore the rank of A is rk(A) = 2.
Exercise 2. (2+1+3 = 6 Points) Let p(x) = a0 + a1 x + a2 x2 + a3 x3 be a polynomial of degree 3 withreal coeﬃcients a0 , a1 , a2 , a3 ∈ R. For such a polynomial p deﬁne the vector vp by a0
a1 4
vp = a2  ∈ R .
a3(i) Find a matrix D ∈ R4×4 , such that vq = Dvp , where q(x) = 2p(x) − p′ (x).
(Here p′ denotes the derivative of the polynomial p with respect to x).
(ii) Determine the rank of D in (i).
(iii) For an arbitrary polynomial p of degree 3, ﬁnd a polynomial s, such that vp = Dvs .

Exercise 3. (2+2 = 4 Points) Let A ∈ R3×3 be a matrix.
(i) Show that if rk(A) = 3 then there exists just one vector x ∈ R3 with Ax = 0.
(ii) Show that if rk(A) ≤ 2 then there exist inﬁnitely many vectors x ∈ R3 with Ax = 0.
(Exercise 4. (4+2 = 6 Points) Let a, b, c, d ∈ R and A =)a b.
c d(i) Show that rk(A) = 2 if and only if ad − bc ̸= 0.
(ii) We deﬁne the following subset of R2
L = {x ∈ R2 | x = Av for some v ∈ R2 }.
How does L look like if rk(A) = 1? How does it look like if rk(A) = 2?

Version: October 13, 20231,;Hello ∼ クマ先生 here. With another homework comes another Japanese lesson∼
This week, I have three words prepared for this time:
ぎょうれつ

ほうていしき

行列

方程式

ベクトル

These three words are: gyouretsu (matrix), houteishiki (equation), and bekutoru (vector). While theJapanese word for vector is a direct transliteration of the English word vector, the other two are not so.
ぎょうれつThe ﬁrst word ( 行 列 ) also has a meaning in everyday life: a procession (e.g. a wedding procession).
ほうていしきThe second word (方程式), however, appears more in the sciences. Famous examples include Schrödinger’s
ほうていしきほうていしきEquation (シュレディンガー方程式) and Maxwell’s Equations (マクスウェルの方程式). Generally, howぎょうれつほうていしき

ever, this word can be used to describe any type of equation. For example, 行列方程式 (Matrix Equation),
だいすうほうていしき

べくとるほう ていし き

せんけいほうていしき

代数方程式 (Algebraic Equation), ベクトル方程式 (Vector Equation) and 線型方程式 (Linear Equation).
かんじ

Anyway, now, a breakdown of the individual 漢字 that makes up these two words:
This kanji means ”row(s)”. It refers to how matrices have ”rows”. In fact, theぎょう行ぎょう-いcommon, as it is the kanji used in 行きます (meaning ”to go” in Japanese).

れつ列-This kanji means ”column(s)”. As such, the Japanese word for matrix literallymeans ”rows of columns” or ”columns of rows”, which is an apt description ofwhat a matrix is. In everyday life, this kanji shows up because of its othermeaning : a queue (i.e. a queue in front of a shop).
This kanji means ”direction” or ”way”. It refers to how an equation gives
direction(s) on how the variables are related to one another. This kanji ap-ほう方”row” of a matrix is called a 行 in Japanese. In everyday life, this kanji very-

ほうほうかたpears in a few words in everyday life, including 方法 (method), (あの) 方 (thatあなたperson), and 貴方 (you; kanji literally means ”precious person”).
てい程

ほうていしき-This kanji means ”formula” or ”ceremony”. This kanji can be found in any

しき式This kanji means ”about”. In a sense, 方程式 (Equation) tells us something
about some natural phenomenon. This kanji is uncommon in everyday life.
そつぎょうしき-にゅうがくしき

ceremony in everyday life, such as 卒 業 式 (Graduation Ceremony), 入 学 式
けっこんしき(Entrance ceremony), and 結婚式 (Wedding ceremony), among others.

And that’s it for today’s (Mathematical) Japanese word(s). またね ∼2