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

Instructor: Henrik Bachmann
Teaching assistant: Yuichiro Toma

Homework 5: Subspaces•Deadline: 17th December, 2023

Exercise 1. (5+3 = 8 Points)(i) Which of the following subsets are subspaces? Justify your answers.
}{
U1 = x ∈ R3 | x21 + x22 = x33 ,{}U2 = x ∈ R4 | x1 − x2 = 2x3 and x2 + x3 = −x4 ,U3 = {x ∈ Rn | Ax = x} ,where A ∈ Rn×n is a ﬁxed matrix,

U4 = {x ∈ R | x • v = 0} ,{}U 5 = x ∈ R2 | x 1 ≤ x 2 .

for a ﬁxed v ∈ Rn ,n(ii) For each subset U in (i) that is a subspace, ﬁnd numbers a, b ≥ 1 and a linear map F : Ra → Rb
such that ker(F ) = U .

Exercise 2. (2+3+3 = 8 Points) Consider the following subspace{}

W = ker(Pu ) = x ∈ R3 | Pu (x) = 0 ,1where u = −1 .
1(i) Determine vectors v1 , . . . , vm ∈ R3 with W = span{v1 , . . . , vm }.
(ii) Find a linear map H : R3 −→ R3 with im(H) = W .
(iii) Calculate ker(H) and ker(Pu ◦ H).

Exercise 3. (3+3 = 6 Points)(i) Let U, V ⊂ Rm be subspaces. Decide whether the union U ∪ V is also a subspace or not.
(ii) Let U, V ⊂ Rm be subspaces. Decide whether the intersection U ∩ V is also a subspace or not.

Version: December 1, 2023

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;The days are getting short, and the nights are getting cold (in Nagoya). Winter has come.
I hope you guys are ready for Professor Bachmann’s Linear Algebra Chrismath Challenge!
Today’s words are mostly on the subject of the HW: Inverses and Subspaces.
かくぞう

か ぎゃくぶ ぶんくうかん

核像

可逆部分空間

These words are: kaku (kernel), zou (image), kagyaku (invertible), and bubun kuukan (subspace).
Here’s a question: how do we say ”invertible matrix” in Japanese?
While I won’t give the answer here, interestingly, there are a few Japanese synonyms for ”invertible
せいそくぎょうれつ

ひとくいぎょうれつmatrix”, namely 正 則 行 列 (lit. regular matrix) and 非特異行列 (lit. non-singular matrix).
ぞう

Note that this time, 像 is omitted due to its inclusion in HW3’s part.
かく核-きょか--ぎゃくThis kanji means ”part”. This word is used in 学部 (Undergraduate). Otherぶ

かぜんぶ

へ やuses of this kanji include 部下 (subordinate), 全部 (all), and 部屋 (room).
This kanji means ”separate”. One word that you might have encountered
わ-that uses this kanji is 分かる (to understand). Other common uses include
きぶん

たぶん気分 (feeling) and 多分 (probably).

This kanji means ”space” or ”sky”. It refers to how subspaces are... spaces!
そら-This kanji is very common in everyday life, with its common uses including 空くうこうくうはくからて(sky), 空港 (airport), 空白 (blank space) and 空手.
This kanji means ”interval”. It refers to how subspaces are ”intervals” ofかん間This kanji means ”reverse”. It refers to how an inverse matrix is a reverse of
the original matrix. One use of this kanji in everyday life is in the expression

がくぶくう空ふ か逆 ですよ (meaning, no, it’s the opposite!).

ぶん分”able” to be inverted. Common uses of this kanji include 可能性 (Possibility),許可 (permission), and 不可 (unable, or ”to fail” a course)

ぶ部

かくへいき

かのうせい

-ぎゃく

逆げんしかくinclude 原子核 (Atomic Nucleus) and 核兵器 (Nuclear Weapons).
This kanji means ”able” or ”consent”. It refers to how an invertible matrix is

か可Read as ”かく”. This kanji usually means ”nuclear” or ”nucleus”. Uses

じかん

-space. This kanji is also common in everyday life, used in (among others) 時間
あいだしゅんかんにんげん(time), 間 (between), 瞬 間 (moment), and 人間 (human).

And that’s it for today. Take care, Good Luck, and Have fun (GLHF)!
かぎゃくぎょうれつ

P.S. In case you didn’t guess it yet, ”Invertible Matrix” is 可 逆 行 列 (kagyaku gyouretsu).
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