[선형대수학] 가역성, 가역행렬 (Invertibility, Inverse Matrix)

$\underline{Def}$

Let $T \colon V \rightarrow W$ be linear, where $V,W$ are vector spaces over $\mathbb{F}$.

$(i)$ A function $U \colon W \rightarrow V$ is an inverse $T$ if $TU=I_{W},\,UT=I_{V}$.

$(ii)$ $T$ is invertible if it has an inverse.

$(iii)$ Such an inverse $U$ of $T$ is unique if ie exists. In this case, we write $U=T^{-1}.$

 

$\underline{Facts}$

Let $T \colon V \rightarrow W,\, U \colon W \rightarrow Z$ be invertible. Then

$(i)$ $(UT)^{-1}=T^{-1}U^{-1}$

$(ii)$ $(T^{-1})^{-1}=T$

 

$\underline{Thm\, revisited}$

Let $V,W$ be finite-dimensional vector spaces with equal dimension. Let $T \colon W \rightarrow W$ be linear.

Then $T$ is invertible $\Leftrightarrow$ $rank(T)=dim(V)$.

 

$\underline{Thm}$

Let $T \colon V \rightarrow W$ be linear & invertible. Then $T^{-1}$ is also linear.

 

$\underline{Proof}$

Let $y_{1},y_{2} \in W,\, c \in \mathbb{F}$.

Since $T$ is 1-1 & onto, $\exists ! x_{1},x_{2} \in V \, s.t. \, T(x_{i})=y_{i}\,for=1,2,$.

Thus,

$$\begin{equation} \begin{split} T^{-1}(cy_{1}+y_{2}) & = T^{-1}(cT(x_{1})+T(x_{2})) \\ & = T^{-1}(T(cx_{1}+x_{2})) \\ & = cx_{1}+x_{2} \\ & = cT{-1}(y_{1})+T^{-1}(y_{2}) \end{split} \end{equation}$$

 

$\underline{Def}$

Let $A \in M_{n \times n}$. We say that $A$ is invertible if $\exists B \in M_{n \times n}$ s.t.

$$AB=BA=I$$

 

$\underline{Rmk}$

In this case, if $\exists C \in M_{n \times n}$ with $AC=CA=I$, then

$$C=CI=C(AB)=(CA)B=IB=B$$

The unique matrix $B$ is called the inverse of $A$ & denoted by $B=A^{-1}$

 

$\underline{Ex}$

$$\begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix}^{-1} = \begin{bmatrix} 3 & -7 \\ -2 & 5 \end{bmatrix}$$