hce_nthu
111年
資訊科學
第 12 題
Let $S$ be the set of all positive real numbers. Now, we want to make $S$ as a vector space in $V$ by asking the following definitions for vectors, vector addition, and scalar multiplication:
• Each element of $S$ will be considered as a "vector" in $V$.
• For $A, B \in S$, a "vector sum" is defined as
$A + B \equiv A B$,
where the product on the right is the usual product of two real numbers.
• For $c \in \mathbb{R}$ (real number), and $A \in S$, a "scalar multiplication" is defined as
$c \cdot A \equiv A^c$,
that is the real number $A$ raised to the $c$ power.
Based on these, what is the zero vector in $V$?
• Each element of $S$ will be considered as a "vector" in $V$.
• For $A, B \in S$, a "vector sum" is defined as
$A + B \equiv A B$,
where the product on the right is the usual product of two real numbers.
• For $c \in \mathbb{R}$ (real number), and $A \in S$, a "scalar multiplication" is defined as
$c \cdot A \equiv A^c$,
that is the real number $A$ raised to the $c$ power.
Based on these, what is the zero vector in $V$?
- A 0
- B 1
- C 2
- D 3
- E 4
思路引導 VIP
如果我們今天重新定義了「加法」的遊戲規則，使其運算方式與平常熟悉的算術完全不同，那麼你會如何運用「單位元（Identity element）」的概念，去尋找那個在這種新規則下，與任何元素運算後都能讓該元素維持原狀的特殊對象呢？
🤖
AI 詳解
AI 專屬家教
恭喜你準確地判斷出正確答案！這顯示你已經跳脫了對「零」這個數字的直觀刻板印象，真正掌握了線性代數中抽象向量空間（Vector Space）的本質。
零向量的定義與運算驗證
在一個向量空間中，所謂的「零向量」 $\mathbf{0}_V$ 並非指數字 $0$，而是必須滿足加法單位元的公理（Axiom）：對於集合中的任意元素 $A$，必須符合 $A + \mathbf{0}_V = A$。在本題特殊的運算規則下，向量加法被定義為數值的「乘積」，因此關係式變成了 $A \cdot \mathbf{0}_V = A$。在正實數集合 $S$ 中，要使乘法運算後保持原值，該元素顯然必須是 $1$。我們可以進一步透過係數積規則驗證：當純量為 $0$ 時，$0 \cdot A = A^0 = 1$，這正符合向量空間中 $0 \mathbf{v} = \mathbf{0}_V$ 的基本性質。
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