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Suppose $T: V \rightarrow W$ is a linear transformation, where $v$ and $W$ are finite-dimensional vector spaces. If $T^*$ is the adjoint of $T$, show that:
$$\operatorname{rank}(T) =\operatorname{rank}(T^*)$$
MSE:从以下2个等式推出
- $\operatorname{dim} N(T)^{\perp}=\operatorname{dim} V - \operatorname{dim}N(T) =\operatorname{dim} R(T)$.
- $R(T^*)^{\perp}=N(T).$
2的证明:
$\forall x\in R(T^*)^{\perp}:\forall y\in W:0=\langle x,T^*y\rangle=\langle Tx,y\rangle\Rightarrow Tx=0\Rightarrow x\in N(T)$
另一方面,
$\forall x\in N(T):\forall y\in W:Tx=0\Rightarrow0=\langle Tx,y\rangle=\langle x,T^*y\rangle\Rightarrow x\in R(T^*)^{\perp}$
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