[선형대수학] 선형 변환이 일대일인 것과 동치 명제들

$\underline{Thm}$

Let $V,W$ be finite-dimensional vector space of equal dimension. Let $T \colon V \rightarrow W$ be linear. Then TFAE:

$(a)$ $T$ is 1-1.

$(b)$ $T$ is onto.

$(c)$ $rank(T)=dim(V)$

 

$\underline{Proof}$

Write $n=dim(V)=dim(W)$.

$(a) \Rightarrow (b)$

Since $T$ is 1-1, by thm, $nullity(T)=0$.

By Dimension Theorem, $n=dim(V)=nullity(T)+rank(T)=0+rank(T)$,and so $dim(R(T))=n$.

Thus, $R(T)=W$.

$(b) \Rightarrow (c)$

Since $T$ is onto, that is, $R(T)=W$, we gave $rank(T)=n=dim(V)$.

$(c) \Rightarrow (a)$

By Dimension Theorem, we have $nullity(T)=0$. By thm, $T$ is 1-1.

 

$\underline{Ex}$

Let $\mathbb{F}^{2} \rightarrow \mathbb{F}^{2}$ be the lindear map by $T(a_{1},a_{2})=(a_{1}+a_{2},a_{1})$.

If $(a_{1},a_{2}) \in N(T)$, then $a_{1}=a_{2}=0$, that is, $(a_{1},a_{2})=0$.

So, $N(T)= \{ 0 \}$. By thm, $T$ is 1-1. So, by above thm, $T$ is also onto.