A nenoaiu o ka laina algebraic nā helu kaulike. Homogeneous nenoaiu o ka laina algebraic nā helu kaulike

Ma ke kula, kela mea keia mea o ko mākou aʻo i ka helu kaulike a me ka, oiaio, o ka nenoaiu o nā helu kaulike. Akā,ʻaʻole he nui na kanaka i ike i loaʻa i kekahi mau aoao, e ke hoʻoponopono iā lākou. I kēia lā eʻike mākou pono a pau i nā kiʻina hana no ka hoʻoponopono i kekahi ano o ka laina algebraic nā helu kaulike, a i haku o ka oi ma mua o nā nā helu kaulike.

moʻolelo

I kēia manawa, ua ike no ia i ka nō o ka hoʻoponopono nā helu kaulike a me kā lākou mau nenoai originated i kahiko Babulona a me Aigupita. Eia naʻe, e hoʻokolokolo pono i loko o kā lākou nā palapala ikea ia mai ma hope o ka hanana o ka like hōʻailona "=", i i hoʻolauna i ka 1556 ma ka 'ōlelo Pelekānia mathematician mooolelo. Ma ke ala, i kēia hōʻailona i waeia no ke kumu: ka mea, 'o ia hoʻi elua laua like Hoʻohana. Aia hoʻi, i ka mea maikai laʻana o ka like, aole ia i hele mai.

I ka mea hoʻoheheʻe kālā o kēia lettering a me ka hōʻailona o ka ike ole loa, i ka Palani mathematician Fransua Viet. Eia naʻe, kona inoa mea me ka nui loa okoa mai i keia la. No ka laʻana, he huinahalike o ka ike heluʻo ia i hoʻokohu 'ia e ka palapala Q (lat "quadratus".), A me ke kupa - (. Lat "cubus") ka palapala C. Kēia mau hōʻailona kēia manawa he koikoi, akā, a laila ka mea, ua ka loa intuitive ala e kākau i kekahi ano o ka laina algebraic nā helu kaulike.

Eia naʻe, he keakea i loko o ka maʻamau ki ina hana like o ka pāʻoihana ua mathematicians i ua manao wale i ka maikaʻi aʻa. Paha keia mea ku pono i ka mea i io Hawaiʻi aʻole 'oe i kekahi e pili ana. i kekahi aoao ai kekahi, akā, me ka mea mua e noonoo io aa hoomaka ma hope o ka Italia makemakika Niccolo Tartaglia, Gerolamo Cardano a me Raphael Bombelli ma ka 16th kenekulia. A kēia e nānā, i ka papa kuhikuhi ano o ka hoʻoponopono quadratic nā helu kaulike (ma discriminant) Ua hoʻokumu wale nō i loko o ka 17th kenekulia ma na hana o Descartes a me Newton.

Ma ka waena o ka 18th kenekulia Hawaiian mathematician Gabriel Cramer loaʻa i ka hou aku e hana i ka pāʻoihana o nenoai o ka laina kaulike e hiki pono. Kēia hana i hope ka inoa ma hope ona, a hiki i kēia lā, ua hana ia. Akā, ma ke ano o ka laua kamailio ana Kramer ka he uuku hope, akā, no ka manawa, e kūkākūkā mākou laina nā helu kaulike a me kā lākou kākoʻo pāʻoihana kaawale mai ke kahua.

laina nā helu kaulike

Laina nā helu kaulike - ka simplest helu kaulike me ka ee iaaanu aey (mau). Ka mea, e pili ana i ka algebraic. Laina kaulike e kākau i loko o ka mau ano like penei: he ₁ * m ₁ + he _{2 *} m ₂ + ... a me _N * m _N = b. Hoolohe ana i keia palapala mākou, e pono i ka makaukau ana o nenoai, a matrices ia.

A nenoaiu o ka laina algebraic nā helu kaulike

Ka wehewehena o keia makahiki mea: he lākou o nā helu kaulike i i ole unknowns, a me ka mau pāʻoihana. Nō kāu, i ke kula a pau Wehewehe i nā hāʻina i ka'ōnaehana me elua paha a ekolu nā helu kaulike. Akā, he mau nenoai me ka eha a oi eiiiiiaiou. E ka ike mua pehea e kākau ia i lalo, no laila, i ka hope ka mea, ua pono ia ke hoʻoponopono. Mua, o ka'ōnaehana o ka laina algebraic nā helu kaulike, e nana maikaʻi ina e kākau i nā aiaiiuo a pau e like me ka m me ka ~ AYEIE inideka: 1,2,3, a no laila, ma luna o. Ka lua, ka mea e alakai i na nā helu kaulike ma ka canonical palapala: he ₁ * m ₁ + he _{2 *} m ₂ + ... a me _N * m _N = b.

Ma hope o nā mea a pau kēia mauʻanuʻu, ua hiki hoʻomaka e hai aku ia oe pehea e huli i ka pāʻoihana o nenoai o ka laina nā helu kaulike. He nui loa no ka mea, e hele mai i loko o mākaukau mua.

'ōpū mai

Mua - ka papaʻaina i ka he o ka lalani a me kolamu, a me kona oihana mua i ma kā lākou kuʻina. Kēia hiki e kekahi i kekahi waiwai a ee iaaanu aey. Ma ka hapanui o na, e hoʻokohu 'ia oihana mua i i ka hooponopono ia ma lalo o nā subscripts (e laʻa me, he ₁₁ a ₂₃ luawai). Ke 'inideka mua Hōʻike ka lalani helu, a me ka lua - ke kolamu. Luna matrices like ma luna a me kekahi'ē aʻe makemakika hehee ai ke hana likeʻole hana. Penei, e hiki:

1) unuhi, a hoʻokomo i ka ia nui o ka papaʻaina.

2) E hoonui i ka'ōpū mai i kekahi helu a maʻi lele.

3) Transpose: hoʻololi kēia hiapo a laina i loko o na kolamu, a me na kolamu - ma ka laina.

4) hoonui i ka mua, ina ka helu o ka lalani Ua like ia me kekahi o ia mau mea he okoa ka helu o nā kolamu.

E kūkākūkā i au mamuli a pau o kēia mau 'ana i kūpono, e like me ka mea, e pono iā mākou i loko o ka wā e hiki mai. Subtraction a me ka hou ana o matrices ka loa mea. Mai ke lawe i ka ia nui mua, kēlā hehee ai o kekahi papa o nā i kela 'ē hehee ai. Pēlā mākou hui (unuhi) mau o kēia mau kumu mua (ia mea nui i ka mea i ku ma ka ia lepo i loko o kā lākou mau matrices). I ka hoonuiia e ka helu o ka'ōpū mai a maʻi lele oe wale hoonui kēlā me kēia hehee ai o ka'ōpū mai ma ia helu (a maʻi lele). Transposition - he loa hana hoihoi kaʻina. Loa hana hoihoi 'kekahi manawa, eʻike iā ia i loko o maoli ola, no ka laʻana, ka wā hoʻololi i ke kohoʻia o ka papa' ole kelepona. Nā icons ma kaʻoneki o ka mua, a me ka loli o ke kulana, ka mea, ua transposed a lilo lehulehu, akā, emi ma kona kiʻekiʻe.

E mākou huli hou i ke kaʻina hana e like me kēia hiapo a multiplication. ʻOiaiʻo ia i haʻi aku iā mākou, a me ka mea i pono keia mau mea, akā, e makaala ia mea nō maikaʻi kēia. E hoonui i mau matrices hiki ia ma lalo o ke kūlana wale i ka helu o ke kolamu i loko o kekahi papa, ua like me ka helu o ka lalani 'ē aʻe. Ano lawe kekahi'ōpū mai laina oihana mua a me nā oihana mua o ka AYEIE kolamu. Hoonui ia i kēlā me kēia huina'ē aʻe, a laila, (a laila, no ka laʻana, he huahana o ka oihana mua ₁₁ a me _12, a ma ₁₂ b. A me ka ₂₂ b. E e like ia: he * b. _{11 12} + ₁₂ * b. A me _22). Pela, ua piha hoi i ka hookahi papaaina'ikamu, a me ka papa hana i like me ia.

Ano, ua hiki hoʻomaka i ka noonoo ana i ka hoʻoponopono i nenoai o ka laina nā helu kaulike.

Gauss

hoomaka kēia theme like e lawe wahi ma ke kula. Ua ike pono no i ka manaʻo o "nenoaiu a mau laina nā helu kaulike" a me ka ike pehea e hoʻoponopono i ia. Akā, i ka mea a inā i ka helu o nā helu kaulike, uaʻoi aku ma mua o elua? Kēia, e kōkua mai Gauss hana.

O ka holo ana, i kēia papa hana mea kupono e hoʻohana, ina oe e ka'ōpū mai o ke kahua. Akā, e hiki ole hoohuli ia, a hooholo ma luna o kona iho.

No laila, pehea e hoʻoponopono i ia ma ke ano o ka laina nā helu kaulike Gauss? Ma ke ala, a hiki nae keia hana, a kapaʻia ma hope ona, akā, loaa ia i loko o ka wa kahiko. Gauss mai he hana lawe mai me na nā helu kaulike, e ho'ōla hopena ma ka totality e echelon ano. Ia mea, e pono i ka luna-ia (ina pololei 'ia o ia wahi), mai ka mua a hiki i ka hope helu kaulike by double kekahi IKE OLE IA. Ma nā hua'ōlelo, ua pono i ka eʻoiaʻiʻo ia makou loaʻa, e olelo aku, ekolu nā helu kaulike: ka mua - unknowns ekolu, i loko o ka lua -ʻelua i loko o ke kolu - hoʻokahi. A laila, mai ka hope helu kaulike, ua loaʻa i ka ike ole ia mua, panai kona waiwai i loko o ka lua a me ka mea mua helu kaulike, a me ka huli hou ka mea i koe mau aiaiiuo.

Cramer ka rula

No ka hooulu ana o kēiaʻano hana mea nui, e ka Haku na ke kūlana mākaukau maoli o ka hou, subtraction o matrices, e like me ka mea pono e e hiki ke loaʻa determinants. No ia mea, ina oe i ke koikoi hana keia a pau a me ka mai i ike pehea, ia mea pono ke aʻo a me ka e hoʻomaʻamaʻa.

He aha ka ka pā like o keia hana, a pehea e hana ai, e kiʻi i kekahi ano o ka laina nā helu kaulike Cramer? It Aloha loa noonoo ole. E Pono e kūkulu i ka'ōpū mai o ka huahelu (aneane mau) na coefficients o ke ano o ka laina algebraic nā helu kaulike. E hana i kēia, wale lawe i ka helu o ka mea ike ole ia, a ua hoʻoponopono ai i ka papaʻaina i loko o ka mea i ka mea i kākauʻia i loko o ka'ōnaehana. Inā ma mua o ka helu o ka hoailona ia "-", laila mākou e kākau io kaʻi lau waiwai. No laila, ua i ka'ōpū mai mua o ka coefficients o na unknowns,ʻaʻole me ka helu ma hope o ka like hoailona (o ka papa, i ka helu kaulike loaʻa ia e hoemi i ka canonical ano i ka wa a ka akau mea pono i ka helu, a me ka hema - a pau na unknowns me coefficients). A laila, e pono e hana i kekahi mau matrices - kekahi no kēlā me kēia ee iaaanu aey. No keia mea, i loko o ka'ōpū mai mua ua UAIAaIN ma kekahi kolamu kēlā me kēia kolamu helu me ka coefficients ma hope o ka like hoailona. Penei ke kiʻi i kekahi mau matrices, a laila, loaʻa lākou determinants.

Ma hope o mākou i loaʻa i nā qualifiers, he uuku. Mākou i ka loiloi mua mua, a ma laila i kekahi mau loko matrices, a hoʻopili like i kekahi aiaiiuo. E kiʻi i ka'ōnaehana pāʻoihana, ke puunaue i ka determinant o ka kūpono papaʻaina ma luna o ka iniiaiie determinant o ka papaʻaina. Ke kūpono ai ka helu o ka waiwai o kekahi ee iaaanu aey. Like, ua loaʻa i nā unknowns.

nā ano

He nui na ano ma ka mea e loaa i ka pāʻoihana o nenoai o ka laina nā helu kaulike. No ka laʻana, he ai-kapaia Gauss-Ioredane hana, a ua hoʻohana 'ia no ka loaa pāʻoihana o ka'ōnaehana o quadratic nā helu kaulike, a me pili i ka pono o matrices. Aia Ua hoi he Jacobi ano no ka hoʻoponopono i kekahi ano o ka laina algebraic nā helu kaulike. He hiki wawe ka ke hakuloli i nā kamepiula 'ē aʻe, a ua hoʻohana' ia i loko o Me.

huikau hihia

Ka manaʻo pohihihi ana ia lŘlŘ ina ka helu o nā helu kaulike Ua emi ma mua o ka helu ana o aiaiiuo. A laila, ua hiki pono ke olelo e, a me ka'ōnaehana mea kue ( 'o ia hoʻi,ʻaʻohe aa), a me ka helu o kona hoʻoholo mālama i ka infinity. Inā mākou i ka lua o ka hihia - ia mea e pono ke kakau i ka nui pāʻoihana o ka'ōnaehana o ka laina nā helu kaulike. Ka mea, e hoʻokomo i ka liʻiliʻi loa hoʻokahi ee iaaanu aey.

ka hopena

Eia ke hele mai a hiki i ka hopena. E hōʻuluʻulu: mākou i e maopopo i ka'ōnaehana mua, aʻo e huli i ka nui pāʻoihana o ke ano o ka laina nā helu kaulike mea. Eia hou, ua noʻonoʻo'ē aʻe nā koho. E kii mai ana, e hoʻoponopono i nenoai o ka laina nā helu kaulike: Gaussian hoʻopau, a Cramer ka noho alii ana. Mākou kamailio e pili ana i na hihia paʻakikī a me nā 'ano o ka loaa pāʻoihana.

I mea, i kēia pukana Ua nui hou nui, a ina e makemake e ike maikaʻi ia, ke hōʻike iā 'oe e heluhelu hou aku o ka hana kūikawā moʻokalaleo.