Hiòng-liōng khong-kan

**Hiòng-liōng khong-kan** sī sò͘-ha̍k lāi-tóe hō-chò sòaⁿ-sèng tāi-sò͘ ê chı̍t-ki teh gián-kiù ê mi̍h-kiāⁿ.

It-poaⁿ lán teh kóng-·ê kí-hô-ha̍k ê hiòng-liōng, lán nā chim-chiok hun-thiah i-ê sèng-chit, liáu-kái i nn̄g-ê ki-pún ūn-sǹg (ia̍h-tio̍h-sī hiòng-liōng ka-hoat kap sûn-liōng sêng-hoat) kap chı̍t-kóa-á hān-chè, lán tio̍h ē-tàng chhōe-tio̍h chı̍t-thò kong-siat lâi biâu-su̍t chit-ê hō-chò hiòng-liōng khong-kan ê mi̍h-kiāⁿ.

Chiàu án-ne lâi kóng, hiòng-liōng m̄-nā sī kí-hô-ha̍k ê hiòng-liōng nā-tiāⁿ. Put-koán siáⁿ-mi̍h sò͘-ha̍k ê mi̍h, nā kap chit-thò kong-siat ū bā, tio̍h sǹg sī hiòng-liōng. Pí-lūn-kóng, kan-taⁿ ēng si̍t-sò͘ chò hē-sò͘ ê chōe-hāng-sek tio̍h chiâⁿ-chò chı̍t-ê hiòng-liōng khong-kan. Hiòng-liōng khong-kan ū chia-·ê thiu-siōng ê te̍k-sèng, tī sò͘-ha̍k ū chiâⁿ-tōa ê lō͘-iōng.

## Chèng-sek ê tēng-gī

Hiòng-liōng khong-kan sī tùi chı̍t-ê thé  **F** (chhin-chhiūⁿ si̍t-sò͘ chiū-sī chı̍t-ê thé, ho̍k-cha̍p-sò͘ mā sī thé) chiâⁿ-sêng ê chı̍p-ha̍p **V**, jî-chhiáⁿ ài ū nn̄g-hāng ūn-sǹg:
**hiòng-liōng ka-hoat*, ēng **v**+**w** lâi phiau-kì, chia kóng-ê **v** kap **w** lóng sī **V** ê sêng-oân, 
**sûn-liōng sêng-hoat*, ēng *a* **v** lâi phiau-kì, chia kóng-ê *a* sī **F** ê sêng-oân, **v** sī **V** ê sêng-oân,
Chit nn̄g-hāng ūn-sǹg ài chun-chiàu ē-kha cha̍p-tiâu kong-siat ê kui-tēng: 
#Hiòng-liōng ka-hoat ài hip-hong, tio̍h-sī kóng, nā-sī **u** kap **v** lóng sī **V** ê sêng-oân(kì-chò **u**, **v** ∈ **V**), **u**+**v** mā tiāⁿ-tio̍h sī **V** ê sêng-oân (kì-chò **u**+**v** ∈ **V**).
#Sûn-liōng sêng-hoat ài hip-hong, tio̍h-sī kóng, nā-sī *a* ∈ **F**, jî-chhiáⁿ **v** ∈ **V**, *a* **v** tiāⁿ-tio̍h ài sī **V** è sêng-oân.
#Hiòng-liōng ka-hoat ài ū kiat-ha̍p hoat-chek, tio̍h-sī kóng, nā-sī **u**, **v**, **w** ∈ **V**, tiāⁿ-tio̍h **u** + (**v** + **w**) = (**u** + **v**) + **w**.
#Hiòng-liōng ka-hoat ài ū kau-oāⁿ hoat-chek, tio̍h-sī kóng, nā-sī **v**, **w** ∈ **V**, tiāⁿ-tio̍h **v** + **w** = **w** + **v**.
#Hiòng-liōng ka-hoat ài ū lêng-oân, tio̍h-sī kóng, tī **V** lāi-tóe ài ū chı̍t ê **0**, hō-chò lêng-hiòng-liōng, put-koán tó-chı̍t-ê **v** ∈ **V**, tiāⁿ-tio̍h **v** + **0** = **v**.
#Hiòng-liōng ka-hoat ài ū ge̍k-oân, tio̍h-sī kóng, put-koán tó-chı̍t-ê **v** ∈ **V**, lán tio̍h ē-tàng chhōe-tio̍h chı̍t-ê **w** ∈ **V**, hō͘  **v**+**w**=**0**. Chit-ê **w** hō-chò **v** ê ka-hoat ge̍k-oân.
#Sûn-liōng sêng-hoat ài ū kiat-ha̍p hoat-chek, tio̍h-sī kóng, nā-sī *a*, *b* ∈ **F**, **v** ∈ **V**, tiāⁿ-tio̍h *a*(*b* **v**) = (*ab*)**v**.
#Sûn-liōng sêng-hoat ài ū tan-ūi-oân, tio̍h-sī kóng, tī **F** lāi-tóe ài ū chı̍t ê *1*, put-koán tó-chı̍t-ê **v** ∈ **V**, tiāⁿ-tio̍h *1* **v** = **v**.
#Sûn-liōng sêng-hoat tùi hiòng-liōng ka-hoat ài ū hun-phòe hoat-chek, tio̍h-sī kóng, nā-sī *a* ∈ **F**, jî-chhiáⁿ **v**, **w** ∈ **V**, tiāⁿ-tio̍h *a*(**v**+**w**) = *a* **v** + *a* **w**.
#Sûn-liōng sêng-hoat tùi thé ê ka-hoat mā ū hun-phòe hoat-chek, tio̍h-sī kóng, nā-sī *a*, *b* ∈ **F**,  jî-chhiáⁿ **v** ∈ **V**, tiāⁿ-tio̍h (*a*+*b*)**v** = *a* **v** + *b* **v**.

**V** ê sêng-oân hō-chò hiòng-liōng, *F* ê sêng-oân hō-chò sûn-liōng. Tōa-pō͘-hūn ê pān-sè sûn-liōng nā m̄-sī si̍t-sò͘ tio̍h-sī ho̍k-cha̍p-sò͘. Nā beh phoàⁿ-toàn chit-ê mi̍h-kiāⁿ sī-m̄-sī hiòng-liōng khong-kan, lán tio̍h-ài kóng-bêng **F** kap **V**, tēng-gī **V** ê hiòng-liōng ka-hoat kap sûn-liōng sêng-hoat. Téng-koân cha̍p-ê kong-siat sī pit-iàu kap chhiong-hūn ê tiâu-kiāⁿ.

## Ki-pún ê sèng-chit

Tùi chia-ê kong-siat ē-tàng khoàⁿ-chhut chı̍t-kóa-á hiòng-liōng khong-kan ê sèng-chit.

***V** lāi-tóe kan-taⁿ ū chı̍t-ê lêng-hiòng-liōng.Nā-sī kóng **V** lāi-tóe ū nn̄g-ê bô-kâng ê lêng-hiòng-liōng **0**1 kap **0**2, chiàu tē-5-tiâu kóng-siat, put-koán tó-chı̍t-ê **v** ∈ **V**, **v**+**0**1=**v**,  só͘-í **0**2+**0**1=**0**2. Koh-lâi, mā-sī tē-5-tiâu, **v**+**0**2=**v**, só͘-í **0**1+**0**2=**0**1. Chiàu tē-4-tiâu,  **0**2+**0**1=**0**1+**0**2. Só͘-í **0**2=**0**1, khêng-sı̍t **0**1 kap **0**2 tio̍h-sī kâng chı̍t-ê mi̍h-kiāⁿ.
*Put-koán tó-chı̍t-ê **v** ∈ **V**, nā kap **F** ê ka-hoat tan-ūi-oân chò sûn-liōng sêng-hoat, kiat-kó it-tēng sī lêng-hiòng-liōng, ia̍h-tio̍h-sī *0* **v**=**0**.
       **v**+*0* **v** = *1* **v** + *0* **v**     --- tē-8-tiâu
                = (*1*+*0*)**v**       --- tē-10-tiâu
                = *1* **v**              --- **F** ê sèng-chit
                = **v**               --- tē-8-tiâu
Koh-lâi, chiàu tē-5-tiâu, *0* **v** tio̍h-sī lêng-hiòng-liōng. Lêng-hiòng-liōng kan-taⁿ ū chı̍t-ê, só͘-í *0* **v**=**0**.
*Put-koán tó-chı̍t-ê *a* ∈ **F**, *a* **0** = **0**.Nā-kóng *a* sī *0*, téng-koân tio̍h í-keng thó-lūn-kè-·ah. Nā m̄-sī, kun-kù thé ê sèng-chit, *a* ài-ū chı̍t-ê sêng-hoat ge̍k-oân *a*-1 . Put-koán tó-chı̍t-ê **v** ∈ **V**, 
            **v**+*a* **0** = *1* **v**+*a* **0**            --- tē-8-tiâu 
                    = *a*  *a*-1 **v** + *a* **0**     --- **F** ê sèng-chit
                    = *a*(*a*-1 **v**+**0**)       --- tē-9-tiâu
                    = *a* **a**-1 **v**             --- tē-5-tiâu
                    = **v**                   --- tē-7-tâu
Koh-lâi, chiàu tē-5-tiâu, *a* **0** tio̍h-sī lêng-hiòng-liōng. Lêng-hiòng-liōng kan-taⁿ ū chı̍t-ê, só͘-í *a* **0**=**0**.
*Nā-sī *a* **v** = **0**, án-ne m̄-sī *a*=*0*, tio̍h-sī **v**=**0**.
Nā *a* m̄-sī *0*, **v** = *1* **v** = *a*-1 *a* **v** = *a*-1 **0** = **0**. Tī **v** m̄-sī **0** ê pān-sè, nā kóng *a* m̄-sī *0*, chiàu thâu-chêng kóng-·ê, **v** ài sī **0** chiah tio̍h, só͘-í *a* it-tēng ài sī *0*.
*Tī **V** lāi-tóe beh chhōe jīm-hô chı̍t-ê **v** ê ge̍k-oân kan-taⁿ chhōe-ū chit-ê.  Nā-kóng **v** ū nn̄g-ê bô-kâng ê ge̍k-oân **w**1 kap **w**2, chiàu tē-6-tiâu, **v**+**w**1=**v**+**w**2=**0**. Téng-hō nn̄g-pêng lóng kā ka **w**1, **w**1+(**v**+**w**1) = **w**1+(**v**+**w**2). Chiàu tē-3-tiâu kap tē-4-tiâu, **w**1+(**v**+**w**1) = (**v**+**w**1)+**w**2. Só-í **w**1+**0**=**0**+**w**2. Chiàu tē-sì-tiâu kap tē-gō͘-tiâu, **w**1=**w**2, khêng-si̍t **w**1 kap **w**2 tio̍h-sī kâng-chı̍t-ê mi̍h-kiāⁿ.
*Nā-sī *-1* sī **F** lāi-tóe sêng-hoat tan-ūi-oân ê ka-hoat ge̍k-oân, put-koán tó-chı̍t-ê **v** ∈ **V**, *-1* **v** tio̍h-sī **v** ê ka-hoat ge̍k-oân, kán-tan siá-chò **-v**.**v** + **-v** = *1* **v** + *-1* **v** = (*1*+*-1*)**v** = *0* **v** = **0**, só͘-í **-v** tio̍h sī **v** ê ka-hoat ge̍k-oân.

## Lē

*Chū-bêng ê hiòng-liōng khong-kanChū-bêng hiòng-liōng khong-kan ê **V** tio̍h-sī {**0**}, lāi-tóe kan-taⁿ ū chı̍t-ê lêng-hiòng-liōng nā-tiāⁿ. Hiòng-liōng ka-hoat kap sûn-liōng sêng-hoat mā lóng sī chū-bêng ê.
*ThéThé pún-sin mā-sī chı̍t-ê hiòng-liōng khong-kan.  **V** tio̍h sī **F**, hiòng-liōng ka-hoat kap sûn-liōng sêng-hoat tio̍h-sī kap thé goân-tóe ê tēng-gī sio-siâng.
*Chō-phiau khong-kanTio̍h-sī it-poan lán tī kí-hô-ha̍k teh kóng ê hiòng-liōng. 
*Bû-hān chō-phiau khong-kan
*Hêng-lia̍t
*Chōe-hāng-sek