数学外書輪講 I 担当 丹下 基生：研究室 (B715) mail(tange@math.tsukuba.ac.jp）
第 1 回（’18 年 4 月 16 日)———————————————————————————————————————————————

注意-1.
1. １日３人程度
2. セミナーとは、書いてあることをわかりやすく解説をすること．
3. かならずノートを作り、発表のための準備をすること。もし、授業時間前までに準備をしていない場合は、次の人に回します。
4. 出席を取る．
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問題-1-1. [英文和訳]
次の英文を和訳せよ．

(1) Euclidean n-space Rn is defined as the set of all n-tuples (x1 , · · · , xn ) of real numbers xi (a“1-tuple of numbers” is just a number and R1 = R, the set of all real numbers). An element ofRn is often called a point in Rn , and R1 , R2 , R3 are often called the line, the plane, and space,respcetively. If x denotes an element in Rn , then x is an n-tuple of numbers, the ith one ofwhich is denoted xi ; thus we can write
x = (x1 , ..., xn ).
(1)(2) Theorem: There are no four points in the plane such that the distance between each pair is anodd integer.
(2)(3) Theorem: A rectangle R with side lengths 1 and x, where x is irrational, cannot be tiled byfinitely many squares (so that the squares have disjoint interiors and cover all of R).
(2)(4) Suppose that complete bipartite graphs H1 , H2 , · · · , Hm disjointly cover all edges of Kn . Let Xk
and Yk be the color classes of Hk . (The set V(Hk ) = Xk ∪ Yk is not necessarily all of V(Kn ).)We assign an n × n matrix Ak to each graph Hk . The entry of Ak in the ith row and jth columnis1 if i ∈ Xk and j ∈ Yk(k)ai j = 0 otherwise
We claim that each of the matrices Ak has rank 1. This is because all the nonzero rows of Ak
are equal to the same vector, namely, the vector with 1s at positions whose indices belong toYk and with 0s elsewhere.
(2)

問題-1-2. [線形代数の復習](1) 正方行列 A, B に対して rank(A · B) ≤ rank(A) であることを示せ。

(2) ベクトル空間 V の基底の定義をいえ。
(3) ２次元ベクトル空間 V の任意の３つのベクトルは一次従属であることを示せ。
(4) V, W ⊂ Rn は部分ベクトル空間とする。このとき、dim(V + W) = dim(V) + dim(W) − dim(V ∩ W)を示せ。

(5) v1 , v2 · · · , vn ∈ V を dim(V) = n のベクトルとする。このとき、これらが一次独立であれば、基底であることを示せ。
(6) A を n 次正方行列とし、v ∈ Cn とする。このとき、連立方程式 Av = 0 に自明でない解があることと、det(A) = 0 であることは同値であることを示せ。
(7) ３つのベクトル v1 , v2 , v3 のうちどの２つをとっても一次独立であるとき、これらは一次独立か？
(8) d/dt : R[t]2 → R[t]2 を微分写像とする。このとき、線形写像 d/dt を適当な基底によって表現せよ。
1 2 を直交行列によって対角化せよ。
(9) 2 1(10) 2 × 2 正方行列で、対角化できない行列の例をあげよ。
———————————————————————————————————————————————• abstract 抽象的な (Vectors are regarded as abstract mathematical objects with particular properties.)• amount to ということになる (The Itô stochastic integral amounts to an integral with respect to a function.)
stochastic integral (確率積分)• assign : 割り当てる (Each vector v can be assigned coordinates.)• axiomatize 公理化する (The first reaction of mathematicians for those paradox was to ’ axiomatize’ Cantor’s intuitiveset theory.)• anyhow いずれにせよ (You should study diﬀerential geometry anyway.)• be force to do することを強制する (The objective of the flight control system is to force the missile to achieve hesteering commands.)• bipartite graph 二部グラフ• call A B : A のことを B とよぶ (We call those number real number.)• claim 主張する (Newton claimed that he invented diﬀerential first )• class: 類 (Equivalent class)• column 列 (Column vectors)• complete graph 完全グラフ• condition 条件 (Necessary and suﬃcient condition.)• constantly いつも、絶えず (For real number x, x2 ≥ 0 is constantly true.)• contain 含む (This set contains any positive integer.)• cover : 覆う (The family of the intervals covers the real line.)• customary 慣例の、慣例である (It is customary in mathematics to write the equation above as follows.)• deal with ... 扱う (In this chapter we deal with a set of linear equations.)• define A as B : A のことを B と定義する．(We define limit of a function as follows.)• A denote B : A は B のことを示す (V denoted a vector space.)• disjointly : ばらばらに (共通部分なく) (These sets are embedded in the space disjointly.)• distance : 距離 (The distance between the two points is 2.)• edge 辺 (We call the set edge.)• element : 要素、元 (Any element in the set A is also contained in B.)• entry : 行列の各成分 (Any entry of the matrix is positive real number.)• exclude 除外する (If several exceptions are excluded, this theorem may be true.)• exercise 練習問題 (This is an easy exercise.)• finitely many : 有限個の (The set contains finitely many elements.)• for instance 例えば (For instance, the equation satisfies with the property.)• a fortiori なおさら (Complex number satisfies it, a fortiori, so is real number.)• fraction 分数 (Partial fraction decomposition)• function 関数 (Continuous function.)• graph グラフ (The graph theory.)• indeed 実際 (Indeed, the computation is a hard work.)• induction 帰納法 (The formula is proven by induction.)• infinitely many : 無限個の (The linear equation admits infinitely many solutions.)• in practice 実際には (In practice, it is diﬃcult to prove the conjecture.)• integer: 整数 (Integer valued-polynomial.)• interior : 内部 (The interior of the set is an open set.)• irrational: 無理数 (Irrational number is the complement of the rational numbers.)• Let A be B. : A を B とする (Let Ax = b be a linear equation over a field k.)• matrix 行列 (複数形は matrices)(The matrix representation.)• namelyつまり (Namely, this example gives an counterexample of this conjecture.)• not necessarily とは限らない (The solutions of the equation are not necessarily positive real numbers.)• n-tuple: n 組み (The n-tuple of real numbers constructs a vector in Rn .)• observe 観察する (よく見る)(Observing this formula, we recall an identity of trigonometric functions.)• odd : 奇数の (Odd number.)• often しばしば (We often omit the notation.)• pair : 組み (The pair is called a twin prime.)• parenthesis (まる) かっこ (We need start from the innermost calculation surrounded by parentheses.)• property 性質 (Property X is understood as follows.)• rank ランク (Rank formula.)• rational : 有理数 (A is a rational.)• rectangle: 長方形 (A rectangle is a quadrilateral with four right angles)• respectively それぞれ (The two angles are 90◦ and 60◦ respectively.)• restrict attention to ....に注意を制限する (The readers should restrict attention to the understanding of this formula.)• row: 行 (This vector is the first row of the matrix.)• set : 集合 (The set contains at least one element.)• A so that B : B となるような A(There exists functions f1 (t) and f2 (t) so that f (t) = f1 (t) + f2 (t), where f1 (t) and f2 (t)are an odd function and an even function respectively.)• square : 正方形 (The diagonals of any square are equal.)• A such that B : B を満たす A(There exists an integer n such that x = 2n + 1.)• suppose: 仮定する．(Suppose that the set is not empty.)• take into account ... を考慮する (If the assumption is taken into account, the proposition is false.)• theorem : 定理 (Fermat’s Last Theorem.)• There exists A: A は存在する (There exists an integer n such that for any real number r with r > n.)• tile : 敷き詰める (Tiling the plane with triangles.)• vector ベクトル (The vector space is a set.)• vector space V over K K 上のベクトル空間 V(Let V be a vector space over K.)• verify 確認する (We need verify that the solution satisfies the equation.)• vertex 頂点 (A rectangle has four vertices.)• where A, ただし A であるとする．(The number i is a complex number, where i is the imaginary unit.)• with A : A の条件のある (The equation at + b = 0 with a even.)
東大の川東先生の「セミナーの準備のしかたについて」(省略)Web 上の川東先生のページを見ること。
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