关于直和分解的专题讨论

$\bf命题：$设$A \in {M_n}\left( F \right)$，证明：${F^n} = \text{Ker}\left( A \right) \oplus \text{Ker}\left( {E - A} \right)$当且仅当$A$为幂等阵

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$\bf命题：$设$\sigma  \in L\left( {V,n,F} \right),f\left( x \right),g\left( x \right) \in F\left[ x \right]$，且$h\left( x \right) = f\left( x \right)g\left( x \right),\left( {f\left( x \right),g\left( x \right)} \right) = 1$，证明：$$Kerh\left( \sigma  \right) =  Kerf\left( \sigma  \right) \oplus Kerg\left( \sigma  \right)$$

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$\bf命题：$设$A,B,C,D \in L\left( V \right)$且两两可交换，$AC + BD = E$，证明：$KerAB = KerA \oplus KerB$

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$\bf命题：$设$\sigma  \in L\left( {V,n,F} \right),V = {V_1} \oplus {V_2}$，证明：$V = \sigma \left( {{V_1}} \right) \oplus \sigma \left( {{V_2}} \right)$当且仅当$\sigma $可逆

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$\bf命题：$$\bf(02浙大九)$设$\sigma  \in L\left( {V,n,F} \right)$，则$V = \operatorname{Im} \left( \sigma  \right) \oplus {\text{Ker}}\left( \sigma  \right) \Leftrightarrow r\left( {{\sigma ^2}} \right) = r\left( \sigma  \right)$

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$\bf命题：$设$V$是数域$P$上的$n$维线性空间，$\sigma$是$V$上的线性变换，证明：$V=Im\sigma\oplus Ker\sigma$当且仅当$Im\sigma=Im\sigma^{2}$

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$\bf命题：$设$V$为实数域上$n$维线性空间，$f$为$V$上的正定对称双线性函数，$U$是$V$的有限子空间，$W = \left\{ {c \in V|f\left( {c,b} \right) = 0,\forall b \in U} \right\}$，证明：$V = U \oplus W$

练习题

附录

$\bf命题：$设$A \in {M_n}\left( F \right)$，则下列命题等价

$(1)$${F^n}{\rm{ = }}N\left( A \right) \oplus R\left( A \right)$   $(2)$$N\left( A \right) \cap R\left( A \right) = \left\{ 0 \right\}$

$(3)$$N\left( {{A^2}} \right) = N\left( A \right)$   $(4)$$r\left( {{A^2}} \right) = r\left( A \right)$   $(5)$$R\left( {{A^2}} \right) = R\left( A \right)$

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$\bf命题：$设$V$是数域$P$上的$n$维线性空间，$\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$是$V$的一组基，令${V_1} = L\left( {{\alpha _1} + {\alpha _2} +  \cdots  + {\alpha _n}} \right)$，且$$V_{2}=\{k_{1}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{n}\alpha_{n}|\sum\limits_{i=1}^{n}k_{i}=0,k_{i}\in P\}$$

证明：$V_{2}$是$V$的子空间，且$V=V_{1}\oplus V_{2}$

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$\bf命题：$设$A \in {M_n}\left( F \right)$，且$A = \left( {\begin{array}{*{20}{c}}{{A_1}}\\{{A_2}}\end{array}} \right)$，证明：${F^n} = Ker\left( {{A_1}} \right) \oplus Ker\left( {{A_2}} \right)$当且仅当$A$为可逆阵

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$\bf命题：$设${V_1} = \left\{ {\left. {A \in {M_n}\left( F \right)} \right|{A^T} = A} \right\},{V_2} = \left\{ {\left. {A \in {M_n}\left( F \right)} \right|{A^T} =  - A} \right\}$，证明：${M_n}\left( F \right) = {V_1} \oplus {V_2}$

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$\bf命题：$设${V_1} = \left\{ {\left. {A \in {M_n}\left( F \right)} \right|tr\left( A \right) = 0} \right\},{V_2} = \left\{ {\left. {kE} \right|k \in F} \right\}$，证明：${M_n}\left( F \right) = {V_1} \oplus {V_2}$

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