关于直和分解的专题讨论

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

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$f命题：$设$sigma  in Lleft( {V,n,F} ight),fleft( x ight),gleft( x ight) in Fleft[ x ight]$，且$hleft( x ight) = fleft( x ight)gleft( x ight),left( {fleft( x ight),gleft( x ight)} ight) = 1$，证明：$$Kerhleft( sigma   ight) =  Kerfleft( sigma   ight) oplus Kergleft( sigma   ight)$$

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$f命题：$设$A,B,C,D in Lleft( V ight)$且两两可交换，$AC + BD = E$，证明：$KerAB = KerA oplus KerB$

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$f命题：$设$sigma  in Lleft( {V,n,F} ight),V = {V_1} oplus {V_2}$，证明：$V = sigma left( {{V_1}} ight) oplus sigma left( {{V_2}} ight)$当且仅当$sigma $可逆

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$f命题：$$f(02浙大九)$设$sigma  in Lleft( {V,n,F} ight)$，则$V = operatorname{Im} left( sigma   ight) oplus { ext{Ker}}left( sigma   ight) Leftrightarrow rleft( {{sigma ^2}} ight) = rleft( sigma   ight)$

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$f命题：$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：$V=Imsigmaoplus Kersigma$当且仅当$Imsigma=Imsigma^{2}$

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$f命题：$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，$W = left{ {c in V|fleft( {c,b} ight) = 0,forall b in U} ight}$，证明：$V = U oplus W$

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附录

$f命题：$设$A in {M_n}left( F ight)$，则下列命题等价

$(1)$${F^n}{ m{ = }}Nleft( A ight) oplus Rleft( A ight)$   $(2)$$Nleft( A ight) cap Rleft( A ight) = left{ 0 ight}$

$(3)$$Nleft( {{A^2}} ight) = Nleft( A ight)$   $(4)$$rleft( {{A^2}} ight) = rleft( A ight)$   $(5)$$Rleft( {{A^2}} ight) = Rleft( A ight)$

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$f命题：$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${V_1} = Lleft( {{alpha _1} + {alpha _2} +  cdots  + {alpha _n}} ight)$，且$$V_{2}={k_{1}alpha_{1}+k_{2}alpha_{2}+cdots+k_{n}alpha_{n}|sumlimits_{i=1}^{n}k_{i}=0,k_{i}in P}$$

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

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

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

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$f命题：$设${V_1} = left{ {left. {A in {M_n}left( F ight)} ight|trleft( A ight) = 0} ight},{V_2} = left{ {left. {kE} ight|k in F} ight}$，证明：${M_n}left( F ight) = {V_1} oplus {V_2}$

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