Cramer ka rula a me kona noi

Cramer ka rula - o kekahi o ka kiko'ī epekema no ka hoʻoponopono nenoai o ka laina algebraic nā helu kaulike (Slough). Kona pololei aie i ka hoʻohana 'ana i nā determinants o ka'ōnaehana mua, e like me ka pono i kekahi o na kapu kau i loko o ka hooiaio ana o ka theorem.

A nenoaiu o ka laina algebraic nā helu kaulike me coefficients no ia, no ka hana, he iiiaanoaa o R - maoli helu o unknowns x1, X2, ..., xn mea he ohi no ka 'aoʻao

ai2 x1 + ai2 X2 + ... ain A xn =ʻelua me au = 1, 2, ..., ka m, (1)

kahi aij,ʻelua - maoli helu. Kēlā me kēia no kēia, ka 'aoʻao ua kapaia ka laina helu kaulike, aij - coefficients o na unknowns,ʻelua - kuokoa coefficients o nā helu kaulike.

pāʻoihana o (1) haawi ia i ka N-dimensional maʻi lele m ° = (x1 °, X2 °, ..., xn °), ma i hakahaka i loko o ka'ōnaehana no na unknowns x1, X2, ..., xn, kela mea keia mea o ka laina ma ka'ōnaehana i maikaʻi loa helu kaulike .

Ka'ōnaehana Ua kapaia pahuhopu nei ina mea i ma ka liʻiliʻi loa kekahi pāʻoihana, a kue, ina e coincides me ka pāʻoihana 'ia paha o ka nele' ia paha.

Ua pono e hoomanao i ka mea e imi pāʻoihana i nenoai o ka laina nā helu kaulike ka hoʻohana i ke ano o Cramer, i mua nenoai e eʻahā like, a holo nōhie o ia hoʻi, ka ia helu ana o unknowns a me nā helu kaulike i loko o ke kahua.

No laila, e hoʻohana Cramer ka hana, e pono i ka liʻiliʻi loa i mea i ka'ōpū mai,ʻo ke ano o ka laina algebraic nā helu kaulike, a me ka mea, ua puka mai ai. A me ka lua, i maopopo i ua kapaia ka determinant o ka'ōpū mai, a kona mau mākau o computation.

E mākou kuhi i keia ike oe a noho. E Ala E! A laila 'oe i ia wale hoʻopaʻanaʻau nā papakuhikuhi hoomaopopo ana Kramer hana. E Nohie memorization hoʻohana i nā kēia notation:

• Det - i ka papa kuhikuhiE determinant o ka'ōpū mai o ka 'oihana;

• deti - o ka determinant o ka'ōpū mai i loaa mai, mai ka'ōpū mai, iniiaiie o ka'ōnaehana ma ke kūapoʻana i-ko kolamu o ka'ōpū mai a hiki i ke kia ao maʻi lele kona hehee wale i na aoao pono o ka laina algebraic nā helu kaulike;

• N - i ka helu o unknowns a me nā helu kaulike i loko o ke kahua.

A laila, Cramer ka rula computation i-ko ke keʻena xi ( 'olelo = 1, .. N) N-dimensional maʻi lele m hiki ke palapala like

xi = deti / Det, (2).

Ma keia hihia, Det pololei okoa mai Aʻohe.

Ke kū hoʻokahi o ka pāʻoihana o ka'ōnaehana ka wā ka mea, ua Kāʻeokūlani, i hoakaka ia ma ka like ole ana o na kumu nui determinant o ka'ōnaehana i Aʻohe. I ole ia, ina o ka huina o (xi), huinahā, pololei 'oluʻolu, laila SLAE he huinahalike'ōpū mai,ʻo infeasible. Kēia hiki'ākoʻakoʻa i pau ka wā ma ka liʻiliʻi loa kekahi o deti nonzero.

Kumu 1. E hoʻoponopono i ka ekolu-dimensional Lau'ōnaehana ka hoʻohana 'ana Cramer ka haʻilula.
2 x1 + X2 + x3 = 31 4,
5 x1 + X2 + x3 = 2 29,
3 x1 - X2 + x3 = 10.

Olelo hooholo. Mākou e kakau iho i ka'ōpū mai o ka'ōnaehana laina ma ka laina, kahi Ai - o ka i-ko lalani o ka'ōpū mai.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3, -1, 1).
Kolamu noa coefficients b. = (31 October 29).

I ka papa kuhikuhiE'ōnaehana o ka determinant Det
Det = a11 a22 a33 + a12 a23 a31 + a31 a21 a32 - a13 a22 a31 - a11 a32 a23 - a33 a21 a12 = 1 - 20 + 12 - 12 + 2 - 10 = -27.

No ka ho omaulia i ka permutation det1 hoʻohana a11 = B1, a21 = B2, a31 = b3. laila
det1 = B1 a22 a33 + a12 a23 b3 + a31 B2 a32 - a13 a22 b3 - B1 a32 a23 - a33 B2 a12 = ... = -81.

Like, e compute det2 hoʻohana 'hakahaka a12 = B1, a22 = B2, a32 = b3, a, me ka, e hoʻomaulia det3 - a13 = B1, a23 = B2, a33 = b3.
A laila, e hiki ke kaha i det2 = -108, a me ka det3 = - 135.
E like me na nā papakuhikuhi Cramer loaʻa x1 = -81 / (- 27) = 3, X2 = -108 / (- 27) = 4, x3 = -135 / (- 27) = 5.

Pane: m ° = (3,4,5).

E hilinai ana ma ka applicability o keia rula, i ke ano o ka hoʻoponopono Kramer nenoai o ka laina kaulike e hiki ke hoʻohana kahiko o ka'āina, no ka laʻana, e noiʻi ana i ka'ōnaehana i ka hiki helu o nā pāʻoihana ke kaumaha ma muli o ka waiwai o ka aiao ho.

Laʻana 2. No ka hoʻoholo 'ia ma ka aiee o ka aiao ke k like ole | kx - y me - 4 | + | m + HI + 4 | <= 0 i pono kekahi pāʻoihana.

Olelo hooholo.
Kēia like ole, ma ka ho'ākāka 'ana o ka papa Module hiki ke hana wale ina na, ka' aoʻao e 'Aʻohe poʻe. No laila, ua hoemiia keia pilikia i loaa i ka pāʻoihana o ka laina algebraic nā helu kaulike

kx - y me = 4,
m + HI = -4.

Ke kākoʻo pāʻoihana a hiki i keia'ōnaehana wale ina mea o ka papa kuhikuhiE determinant o ka
Det = ke k ^ {2} + 1 mea nonzero. Ua Ua maopopo i keia kulana mea māʻona no a pau maoli aiee o ka aiao ke k.

Pane mai: no ka mea, a pau maoli aiee o ka aiao ke k.

hiki ke hoemi nā Pahuhopu o kēia 'ano i na mea hiki pilikia i loko o ke kula o ka makemakika, ke kālaikūlohea paha kemika.