อ้างอิง:
ข้อความเดิมเขียนโดยคุณ Tohn
2.) Let $V$ and $W$ be vector space over field $F$ and $W \neq \{0\}.$ If $v \in V$ is such that $T(v)=0$ for all $T \in L(V,W),$ prove that $ v=0.$
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มั่วอะครับ ลองเช็คให้ด้วย
Let $v \in V$ be such that $v \neq 0$. Thus $\{ v \}$ is linearly independent.
Then there exist $B \subseteq V$ such that $\{ v \} \subseteq B$ and $B$ is a basis of $V$.
Let $B=\{ u_{i} : i\in I \}$ and $B'=\{ w_{i} : i \in I \} - \{0 \} \subseteq W$.
Then there exist linear transformation $T:V\rightarrow W$ defined by
$$T(u_{i})=w_{i}$$
for all $u_{i} \in B$.
Since $W \neq \{0\}$ for all $i \in I$, $T(u_{i}) \neq 0$.