Sòaⁿ-sèng thí-khui

Hiòng-liōng khong-kan tiong, chi̍t ê hiòng-liōng chi̍p-ha̍p $S$ ê sòaⁿ-sèng thí-khui (Hàn-jī: 線性褫開, Eng-gí: linear span), tiō sī pau-hâm $S$ ê siōng sè chú-khong-kan.

Tēng-gī

Siat-sú ū chi̍t ê tī thé $K$ téng-bīn ê hiòng-liōng khong-kan $V$. Hiòng-liōng chi̍p-ha̍p $S$ ê sòaⁿ-sèng thí-khui (kán-chheng thí-khui) tiō sī só͘-ū pau-hâm $S$ ê chú-khong-kan ê kau-chi̍p $W$. Lán kā siá-chò $W=\operatorname {span} (S)$, mā ē-ēng-tit sú-iōng ē-bīn kúi-nā chióng kóng-hoat:

• $W$ sī hō͘ $S$ thí ·khui ê chú-khong-kan. (Eng-gí: W is the subspace spanned by S.)
• $S$ thí ·khui $W$. (Eng-gí: S spans W.)
• $S$ sī $W$ ê thí-khui chi̍p-ha̍p. (Eng-gí: S is a spanning set of W.)

Lēng-gōa chi̍t chióng tēng-gī sī án-ne: $S$ ê thí-khui tiō-sī só͘-ū $S$ ê goân-sò͘ ê iú-hān sòaⁿ-sèng cho͘-ha̍p, i.e.,

$\operatorname {span} (S)=\left\{{\left.\sum _{i=1}^{n}c_{i}\mathbf {v} _{i}\;\right|\;n\in \mathbb {N} ,\mathbf {v} _{i}\in S,c_{i}\in K}\right\}.$

Lē

• $\left\{(-1,0,0),(0,1,0),(0,0,1)\right\}$ sī ê si̍t-sò͘ hiòng-liōng khong-kan $\mathbb {R} ^{3}$ê chi̍t ê thí-khui chi̍p-ha̍p. Chi̍t ê te̍k-sû ê thí-khui chi̍p mā sī $\mathbb {R} ^{3}$ ê chi̍t ê ki-té. Nā-chún kā $(-1,0,0)$ ōaⁿ-chò $(1,0,0)$, chit ê chi̍p-ha̍p tiō ē chiâⁿ-chò $\mathbb {R} ^{3}$ ê phiau-chún ki-té.

• $\left\{(1,2,3),(0,1,2),\left(-1,{\frac {1}{2}},3\right),(1,1,1)\right\}$ mā sī $\mathbb {R} ^{3}$ê chi̍t ê thí-khui chi̍p-ha̍p, m̄-kò i m̄ sī ki-té, in-ūi i bô sòaⁿ-sèng to̍k-li̍p.
• $\left\{(1,0,0),(0,1,0),(1,1,0)\right\}$ m̄ sī $\mathbb {R} ^{3}$ ê thí-khui chi̍p-ha̍p, tān-sī i thí ·khui $\mathbb {R} ^{2}$.