随便写写的复习笔记Day1，2022/8/22

2022-08-22 06:44 作者:臥煙鈴 0人读过 | 我要投稿

随便写写而已，顺便锻炼一下LateX，如果有写得不对的地方欢迎各位指出，谢谢

刚刚开始就先复习点老得掉牙的题吧

(安徽大学2005)若 $%5Clim_%7Bn%5Cto%20%5Cinfty%7Dx_%7Bn%7D%3Da%2C%5Clim_%7Bn%5Cto%5Cinfty%7D%20y%20_%7Bn%7D%3Db%2C%0A$

证明： $%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7Bx_%7B1%7Dy_%7Bn%7D%2Bx_%7B2%7Dy_%7Bn-1%7D%2B%5Ccdots%2Bx_%7Bn%7Dy_%7B1%7D%7D%7Bn%7D%3Dab$

证:

由 $%5Clim_%7Bn%5Cto%20%5Cinfty%7Dx_%7Bn%7D%3Da%2C%5Clim_%7Bn%5Cto%5Cinfty%7D%20y%20_%7Bn%7D%3Db%0A$ 我们知道 $%5Cforall%20%5Cvarepsilon%20%3E0$ 分别 $%5Cexists%20N_%7B1%7D%EF%BC%8CN_%7B2%7D%3E0%2Cst%20%5Cforall%20n%3EN_%7B1%7D$ 有 $%7Cx_%7Bn%7D-a%7C%3C%5Cvarepsilon$ 且 $%5Cforall%20n%3EN_%7B2%7D%EF%BC%8C%7Cy_%7Bn%7D-b%7C%3C%5Cvarepsilon$

我们不妨令 $N%3DMax%5Cleft%5C%7BN_%7B1%7D%2CN_%7B2%7D%5Cright%5C%7D%2B1$

不妨考虑式子 $%7C%5Cfrac%7Bx_%7B1%7Dy_%7Bn%7D%2Bx_%7B2%7Dy_%7Bn-1%7D%2B%5Ccdots%2Bx_%7Bn%7Dy_%7B1%7D%7D%7Bn%7D-ab%7C$

简单化简有 $%7C%5Cfrac%7Bx_%7B1%7Dy_%7Bn%7D%2Bx_%7B2%7Dy_%7Bn-1%7D%2B%5Ccdots%2Bx_%7Bn%7Dy_%7B1%7D%7D%7Bn%7D-ab%7C$

$%3D%7C%5Cfrac%7B(x_%7B1%7Dy_%7Bn%7D-ab)%2B(x_%7B2%7Dy_%7Bn-1%7D-ab)%2B%5Ccdots%2B(x_%7Bn%7Dy_%7B1%7D-ab)%7D%7Bn%7D%7C$

$%5Cleq%20%0A%7C%5Cfrac%7B(x_%7B1%7Dy_%7Bn%7D-ab)%7D%7Bn%7D%7C%2B%7C%5Cfrac%7B(x_%7B2%7Dy_%7Bn-1%7D-ab)%7D%7Bn%7D%7C%2B%5Ccdots%2B%7C%5Cfrac%7B(x_%7Bn%7Dy_%7B1%7D-ab)%7D%7Bn%7D%7C$ $%5Cleq%0A%5Cfrac%7B%7Cx_%7B1%7D(y_%7Bn%7D-b)%7C%2B%7Cbx_%7B1%7D-ab%7C%7D%7Bn%7D%2B%5Cfrac%7B%7Cx_%7B2%7D(y_%7Bn-1%7D-b)%7C%2B%7Cbx_%7B2%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%5Cfrac%7B%7Cx_%7BN%7D(y_%7Bn-N%2B1%7D-b)%7C%2B%7Cbx_%7BN%7D-ab%7C%7D%7Bn%7D%5C%5C%2B%5Cfrac%7B%7Cx_%7BN%2B1%7D(y_%7Bn-N%7D-b)%7C%2B%7Cbx_%7BN%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%2B%5Cfrac%7B%7Cx_%7Bn-N%2B1%7D(y_%7BN%7D-b)%7C%2B%7Cbx_%7Bn-N%2B1%7D-ab%7C%7D%7Bn%7D%5C%5C%2B%5Cfrac%7B%7Cy_%7BN%2B1%7D(x_%7Bn-N%7D-a)%7C%2B%7Cay_%7BN%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%2B%5Cfrac%7B%7Cy_%7B1%7D(x_%7Bn%7D-a)%7C%2B%7Cay_%7B1%7D-ab%7C%7D%7Bn%7D$ $%3C%0A%5Cfrac%7B%7Cx_%7B1%7D%5Cvarepsilon%7C%2B%7Cbx_%7B1%7D-ab%7C%7D%7Bn%7D%7C%2B%5Cfrac%7B%7Cx_%7B2%7D%5Cvarepsilon%7C%2B%7Cbx_%7B2%7D-ab%7C%7D%7Bn%7D%7C%2B%5Ccdots%2B%5Cfrac%7B%7Cx_%7BN%7D%5Cvarepsilon%7C%2B%7Cbx_%7BN%7D-ab%7C%7D%7Bn%7D%7C%2B%5Cfrac%7B%7Cx_%7BN%2B1%7D%5Cvarepsilon%7C%2B%7Cbx_%7BN%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%5C%5C%2B%5Cfrac%7B%7Cx_%7Bn-N%2B1%7D%5Cvarepsilon%7C%2B%7Cbx_%7Bn-N%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Cfrac%7B%7Cy_%7BN%2B1%7D%5Cvarepsilon%7C%2B%7Cay_%7BN%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%2B%5Cfrac%7B%7Cy_%7B1%7D%5Cvarepsilon%7C%2B%7Cay_%7B1%7D-ab%7C%7D%7Bn%7D$ 这里我们不妨令 $M%3DMax%5Cleft%5C%7B%7Cx_%7Bn%7D%7C%2C%7Cy_%7Bn%7D%7C%5Cright%5C%7D_%7Bn%3D1%2C2%2C%5Cldots%2Cn%7D$

于是

$%7C%5Cfrac%7Bx_%7B1%7Dy_%7Bn%7D%2Bx_%7B2%7Dy_%7Bn-1%7D%2B%5Ccdots%2Bx_%7Bn%7Dy_%7B1%7D%7D%7Bn%7D-ab%7C$

$%3C%0A%5Cfrac%7B%7Cbx_%7B1%7D-ab%7C%7D%7Bn%7D%2B%5Cfrac%7B%7Cbx_%7B2%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%2B%5Cfrac%7B%7Cbx_%7BN%7D-ab%7C%7D%7Bn%7D%2B%5Cfrac%7B%7Cbx_%7BN%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%5C%5C%2B%5Cfrac%7B%7Cbx_%7Bn-N%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Cfrac%7B%7Cay_%7BN%2B1%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%2B%5Cfrac%7B%7Cay_%7B1%7D-ab%7C%7D%7Bn%7D%2B%5Cfrac%7BnM%5Cvarepsilon%7D%7Bn%7D$ $%3C%0A%5Cfrac%7B%7Cbx_%7B1%7D-ab%7C%7D%7Bn%7D%7C%2B%5Cfrac%7B%7Cbx_%7B2%7D-ab%7C%7D%7Bn%7D%7C%2B%5Ccdots%2B%5Cfrac%7B%7Cbx_%7BN-1%7D-ab%7C%7D%7Bn%7D%5C%5C%2B%5Cfrac%7B%7Cay_%7BN-1%7D-ab%7C%7D%7Bn%7D%2B%5Ccdots%2B%5Cfrac%7B%7Cay_%7B1%7D-ab%7C%7D%7Bn%7D%2B%5Cfrac%7BnM%5Cvarepsilon%7D%7Bn%7D%2B%5Cfrac%7Bb(n-2N)%5Cvarepsilon%2B2a%5Cvarepsilon%7D%7Bn%7D$ 又令 $P%3DMax%5Cleft%5C%7B%7Cx_%7Bn%7D-a%7C%2C%7Cy_%7Bn%7D-b%7C%5Cright%5C%7D_%7Bn%3D1%2C2%2C%E2%80%A6%2CN-1%7D$

$%7C%5Cfrac%7Bx_%7B1%7Dy_%7Bn%7D%2Bx_%7B2%7Dy_%7Bn-1%7D%2B%5Ccdots%2Bx_%7Bn%7Dy_%7B1%7D%7D%7Bn%7D-ab%7C$

$%3C%0A%5Cfrac%7BnM%5Cvarepsilon%7D%7Bn%7D%2B%5Cfrac%7Bb(n-2N)%5Cvarepsilon%2B2a%5Cvarepsilon%7D%7Bn%7D%2B%5Cfrac%7B(a%2Bb)P(N-1)%7D%7Bn%7D$

于是便容易得知 $%5Cforall%20%5Cvarepsilon_%7B2%7D%3E0%EF%BC%8C%5Cexists%20G%3E0%2Cst%5C%20%5Cforall%20n%3E0$ 有

成立。

2.设 $dim_%7BF%7DV%3Dn%2C%5Csigma%20%5Cin%20End%20V$ ,证明:

若 $%5Csigma%5E%7Bn-1%7D%5Cneq0%EF%BC%8C%5Csigma%5E%7Bn%7D%3D0%2C$ 则V只有n+1个 $%5Csigma-$ 不变子空间

证:

i)由 $dim_%7BF%7DV%3Dn$ 且 $%5Csigma%5E%7Bn-1%7D%5Cneq0%EF%BC%8C%5Csigma%5E%7Bn%7D%3D0%2C$ 我们得知 $%5Csigma%0A$ 的特征多项式为 $f_%7B%5Csigma%7D(%5Clambda)%3D%5Clambda%5E%7Bn%7D$

于是我们得知 $%5Csigma%0A$ 的所有特征值都是0，而我们容易知道 $ker%5Csigma%3D%5Cleft%5C%7B%5Csigma%5E%7Bn-1%7D%5Calpha%7C%5Cforall%5C%20%5Calpha%20%5Cin%20V%5Cright%5C%7D$ ,我们不妨构造线性空间 $W_i%3D%5Cleft%5C%7B%5Csigma%5E%7Bi%7Dv%7C%5Cforall%20v%5Cin%20V%20%5Cright%5C%7D%2Ci%3D0%2C1%2C%5Cldots%2Cn%EF%BC%8C(W_%7B0%7D%3DV)%0A%E5%85%B6%E4%B8%AD0%5Cin%20W_i$

我们易知 $%5Csigma%5E%7Bm%7Dw%3D0%5Ciff%20w%5Cin%20W_%7Bi%7D%2Ci%5Cgeq%20n-m$ ，任取 $%5Calpha%5Cin%20W_%7Bi%7D$ 于是我们令 $%5Cbeta%3D%5Csigma%5Calpha$ ,则有 $m%5Cgeq%20n-i%EF%BC%8C%5Csigma%5E%7Bm%7D%CE%B2%3D%5Csigma%5E%7Bm%2B1%7D%5Calpha%3D0$ ,

所以线性空间 $W_i%3D%5Cleft%5C%7B%5Csigma%5E%7Bi%7Dv%7C%5Cforall%20v%5Cin%20V%20%5Cright%5C%7D%2Ci%3D0%2C1%2C%5Cldots%2Cn%EF%BC%8C(W_%7B0%7D%3DV)%0A$ 都是 $%5Csigma-$ 不变子空间，而我们也容易知道 $W_n%3CW_%7Bn-1%7D%3C%5Cldots%3CW_1%3CV%0A$

而不妨再假设存在 $W_i$ 以外的不变子空间 $U%5C%20st%20%5C%20%5Cforall%20u%5Cin%20W'%2C%5Csigma%20u%20%5Cin%20W'$ 成立

由 $dim_%7BF%7DV%3Dn$ ，且 $W_n%3CW_%7Bn-1%7D%3C%5Cldots%3CW_1%3CV%0A$ 我们可以知道

$dimW_n%3CdimW_%7Bn-1%7D%3C%5Cldots%3CdimW_1%3CdimV%3Dn%0A$

于是 $dimW_i%3Dn-i%EF%BC%8Ci%3D1%2C2%2C%5Ccdots%2Cn%0A$

由于 $n-1%3DdimW_1%3CdimV%3Dn%0A$

于是我们有 $%5Cforall%20%5Calpha%5Cin%20V%5Ccap%20W_1%EF%BC%8C%5Csigma%5En%5Calpha%3D0%EF%BC%8C%5Csigma%5E%7Bn-1%7D%5Calpha%5Cneq0$

所以不妨令 $U%3D%5BV%5Ccap%20W_1%5D%5Ccup%20%5Cleft%5C%7B0%5Cright%5C%7D$

于是 $dim%20U%3D1%2Cand%20%EF%BC%8C%5Cforall%20%5Calpha%20%5Cin%20U%2C%5Calpha%5Cneq0%2C%5Cforall%20k%5Cin%20F%2C%5Csigma%5Enk%5Calpha%3D0%2C%5Csigma%5E%7Bn-1%7Dk%5Calpha%5Cneq0$

我们不妨取一个 $%20%5Cbeta%20%5Cin%20U$ ，构造向量组 $%5Cbeta%2C%5Csigma%5Cbeta%2C%5Csigma%5E2%5Cbeta%2C%5Cldots%2C%5Csigma%5E%7Bn-1%7D%5Cbeta$

设存在一组数 $k_0%2Ck_1%2C%5Cldots%2Ck_%7Bn-1%7D%5Cin%20F%2Cst%20%5Csum_%7Bi%3D0%7D%5E%7Bn-1%7Dk_i%5Csigma%5E%7Bi%7D%5Cbeta%3D0$

则 $%5Csigma%5Ej%20%5Csum_%7Bi%3D0%7D%5E%7Bn-1%7Dk_i%5Csigma%5E%7Bi%7D%5Cbeta%3D%5Csum_%7Bi%3D0%7D%5E%7Bn-1%7Dk_i%5Csigma%5E%7Bi%2Bj%7D%5Cbeta%3D0%2Cj%3D1%2C2%2C%5Cldots%2Cn-1$

我们可以得到

$%5Csum_%7Bi%3D0%7D%5E%7Bn-j-1%7Dk_i%5Csigma%5E%7Bi%2Bj%7D%5Cbeta%3D0%2Cj%3D1%2C2%2C%5Cldots%2Cn-1$

当 $j%3Dn-1%EF%BC%8C%E6%9C%89%5Csum_%7Bi%3D0%7D%5E%7Bn-j-1%7Dk_i%5Csigma%5E%7Bi%2Bj%7D%5Cbeta%3Dk_0%5Csigma%5E%7Bn-1%7D%5Cbeta%3D0$

当 $j%3Dn-2%EF%BC%8C%E6%9C%89%5Csum_%7Bi%3D0%7D%5E%7Bn-j-1%7Dk_i%5Csigma%5E%7Bi%2Bj%7D%5Cbeta%3D0%2Bk_1%5Csigma%5E%7Bn-1%7D%5Cbeta%3D0$

以此类推得到

$k_0%3Dk_1%3D%5Cldots%3Dk_%7Bn-1%7D%3D0$

所以向量组 $%5Cbeta%2C%5Csigma%5Cbeta%2C%5Csigma%5E2%5Cbeta%2C%5Cldots%2C%5Csigma%5E%7Bn-1%7D%5Cbeta$ 线性无关，所以向量组 $%5Cbeta%2C%5Csigma%5Cbeta%2C%5Csigma%5E2%5Cbeta%2C%5Cldots%2C%5Csigma%5E%7Bn-1%7D%5Cbeta$ 是V的一组基

$%5Cforall%20%5Calpha%20%5Cin%20V%2C%5Cexists%20k_0%2Ck_1%2C%5Cldots%2Ck_%7Bn-1%7D%5Cin%20F%2Cst%5C%20%5Calpha%3D%20%5Csum_%7Bi%3D0%7D%5E%7Bn-1%7Dk_i%5Csigma%5E%7Bi%7D%5Cbeta$

于是我们不妨假设

$%5Cforall%20%5Calpha%20%5Cin%20W'%2C%5Cexists%20k_0%2Ck_1%2C%5Cldots%2Ck_%7Bn-1%7D%5Cin%20F%2Cst%5C%20%5Calpha%3D%20%5Csum_%7Bi%3D0%7D%5E%7Bp%7Dk_i%5Csigma%5E%7Bi%7D%5Cbeta%EF%BC%8Cn-1%5Cgeq%20p%5Cgeq0$

因为 $%5Csigma%20%5Calpha%20%5Cin%20W'$ 所以 $%5Csum_%7Bi%3D0%7D%5E%7Bp%7Dk_i%5Csigma%5E%7Bi%2B1%7D%5Cbeta%5Cin%20W'$ 所以 $%5Csigma%20%5Calpha%2C%5Csigma%5E2%20%5Calpha%2C%5Cldots%2C%5Csigma%5E%7Bn-p-1%7D%20%5Calpha%20%5Cin%20W'$

且我们有 $%5Csigma%5E%7Bm%7D%20%5Calpha%20%3D0%EF%BC%8Cm%5Cgeq%20n-p$ ,所以 $W'%5Csubseteq%20W_p$

而由于 $%5Csum_%7Bi%3D0%7D%5E%7Bp%7Dk_i%5Csigma%5E%7Bi%2Bj%7D%5Cbeta%5Cin%20W'%EF%BC%8C0%5Cleq%20j%5Cleq%20n-p-1$

我们便容易知道 $%5Cbeta%2C%5Csigma%5Cbeta%2C%5Csigma%5E2%5Cbeta%2C%5Cldots%2C%5Csigma%5E%7Bp%7D%5Cbeta%5Cin%20W'$

所以 $span%5Cleft%5C%7B%5Cbeta%2C%5Csigma%5Cbeta%2C%5Csigma%5E2%5Cbeta%2C%5Cldots%2C%5Csigma%5E%7Bp%7D%5Cbeta%5Cright%5C%7D%5Csubseteq%20%20W'$ 于是 $W'%3DW_p$

所以 $W_i%3D%5Cleft%5C%7B%5Csigma%5E%7Bi%7Dv%7C%5Cforall%20v%5Cin%20V%20%5Cright%5C%7D%2Ci%3D0%2C1%2C%5Cldots%2Cn%EF%BC%8C(W_%7B0%7D%3DV)%0A%E5%85%B6%E4%B8%AD0%5Cin%20W_i$ 是所有的 $%5Csigma-$ 不变子空间。

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随便写写的复习笔记Day1，2022/8/22

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