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Assume now that $V$ and $W$ are finite dimensional. Let $\mathcal{B}=\left\{e_1, \cdots, e_n\right\}$ be a basis for $V$ with $\mathcal{E}=\left\{e_1, \cdots, e_k\right\}$ a basis for a subspace $A \subseteq V$ (so $k \leq n$). Let $\mathcal{B}^{\prime}=\left\{e_1^{\prime}, \cdots, e_m^{\prime}\right\}$ be a basis for $W$ with $\mathcal{E}^{\prime}=\left\{e_1^{\prime}, \cdots, e_{\ell}^{\prime}\right\}$ a basis for a subspace $B \subseteq W$. The induced bases for $V / A$ and $W / B$ are given by
$$
\begin{gathered}
\overline{\mathcal{B}}=e_{k+1}+A, \cdots, e_n+A \text { and } \\
\overline{\mathcal{B}^{\prime}}=e_{\ell+1}^{\prime}+B, \cdots, e_m^{\prime}+B .
\end{gathered}
$$
Let $T: V \rightarrow W$ be a linear map such that $T(A) \subseteq B$. Then $T$ induces a map $\bar{T}$ on quotients by Lemma 3.7 and restricts to a linear map
$$
\left.T\right|_A: A \rightarrow B \text { with }\left.T\right|_A(v)=T(v) \text { for } v \in A
$$
Theorem 3.9. There is a block matrix decomposition$$_{\mathcal{B}^{\prime}}[T]_{\mathcal{B}}=\left[\begin{array}{c|c}_{\mathcal{E}^\prime}\left[\left.T\right|_{A}\right]_{\mathcal{E}} & * \\ \hline 0 & _{\overline{\mathcal{B}^{\prime}}}[\bar{T}]_{\overline{\mathcal{B}}}\end{array}\right]$$
$A$为线性空间$V$的子空间,线性变换$T:V\to W$,若$T|_A$和$T|_{V/A}$都可逆,则$T$可逆吗
凑了一下
\[\pmatrix{a&b\\&c}\pmatrix{a^{-1}&-a^{-1}bc^{-1}\\&c^{-1}}=\pmatrix{1\\&1}\]所以$a,c$都可逆可以推出$\pmatrix{a&b\\&c}$可逆吗 |
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