Fesili Fa'ata'ita'i e Talanoaina ai le Sootaga i le va o Matrices ma Suiga
Pendahuluan
O se matrix o se fa'asologa fa'atafafā o numera po'o elemene o lo'o fa'atulagaina i laina ma koluma. E fa'aaogaina lautele matrices i matā'upu eseese e pei o fuainumera, fisiki, tamaoaiga, ae maise lava i suiga fa'ata'ita'i i le matematika ma ata fa'akomepiuta. E maua fo'i e matrices ni meafaigaluega aoga mo le fa'aogaina o fa'amaumauga ma mo le fa'amatalaina ma le fo'ia o fa'afitauli fa'amatematika eseese. O se tasi o fa'aoga taua o matrices o lo'o i suiga fa'asolosolo, lea e fa'aaogaina ai galuega fa'ata'ita'i e suia ai le foliga ma le tulaga o mea fa'ata'ita'i i le avanoa.
I totonu o lenei tusiga, o le a tatou talanoaina ai nisi o faʻataʻitaʻiga o faʻafitauli e faʻaalia ai le faʻaaogaina o matrices mo suiga laina, ma o le a faʻamatalaina auiliili a latou fofo.
Fa'amatalaga ma Fa'amatalaga
Mo se amataga, seʻi o tatou iloiloina nisi o faʻamatalaga ma faʻamatalaga faʻavae o le a faʻaaogaina i lenei talanoaga:
1. Matrix: O se fa'asologa fa'atafafā o numera ua fa'atulagaina i laina ma koluma.
2. Suiga Fa'asolosolo: O se galuega faatino e ave ai se vector ma fa'afanua i se isi vector e fa'aaoga ai galuega fa'atino matrix.
3. Vector: O se elemene o se seti vector e iai le umi ma le itu e agai i ai, e masani ona faʻatusalia o se koluma poʻo se laina i totonu o se matrix.
O fa'ailoga o le matrix e masani ona tusia i mata'itusi tetele, mo se fa'ata'ita'iga \( A \), \( B \), ma o vectors e tusia i mata'itusi tetele pe fa'atasi ai ma se aū i luga a'e, mo se fa'ata'ita'iga \( \mathbf{v} \) po'o le \( \vec{v} \).
Fesili Fa'ata'ita'i ma Talanoaga
Fesili 1: Suiga Fa'ata'amilosaga
I le tuuina atu o se matrix o le suiga o le rotation \( R \) i se tulimanu \( \theta \) i le avanoa e lua-dimensional:
\[ R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \]
Vector \( \mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \). Fuafua le iʻuga o le suiga o le vector \( \mathbf{v} \) e ala i le matrix \( R \) pe afai \( \theta = \frac{\pi}{2} \).
Talanoaga:
Muamua, fa'aofi tau o le tulimanu \( \theta = \frac{\pi}{2} \) i totonu o le matrix \( R \):
\[ R = \begin{pmatrix} \cos\frac{\pi}{2} & -\sin\frac{\pi}{2} \\ \sin\frac{\pi}{2} & \cos\frac{\pi}{2} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]
Sosoo ai, fa'atele le matrix \( R \) i le vector \( \mathbf{v} \):
\[ R \mathbf{v} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} (0 \cdot 1) + (-1 \cdot 0) \\ (1 \cdot 1) + (0 \cdot 0) \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]
O lea la, o le taunuuga o le liua o le vector \( \mathbf{v} \) e le matrix \( R \) mo le tulimanu \( \theta = \frac{\pi}{2} \) o le vector \( \mathbf{v'} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \).
Fesili 2: Suiga o le Fua
I le tuʻuina atu o se matrix o le suiga o le fua \( S \) i le avanoa lua-vaega e pei ona taua i lalo:
\[ S = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \]
Vector \( \mathbf{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \). Saili le i'uga o le suiga o le vector \( \mathbf{u} \) e le matrix \( S \).
Talanoaga:
Fa'atele le matrix \( S \) i le vector \( \mathbf{u} \):
\[ S \mathbf{u} = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} (2 \cdot 1) + (0 \cdot 2) \\ (0 \cdot 1) + (3 \cdot 2) \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} \]
O lea la, o le taunuuga o le liua o le vector \( \mathbf{u} \) e le matrix \( S \) o le vector \( \mathbf{u'} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} \).
Fesili 3: Suiga o le Mafaufau Loloto
I le tuuina atu o le matrix o le atagia \( F \) e faatatau i le y-axis:
\[ F = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \]
Fuafua le i'uga o le liua o le vector \( \mathbf{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \) e fa'aaoga ai le reflection matrix \( F \).
Talanoaga:
Fa'atele le matrix \( F \) i le vector \( \mathbf{w} \):
\[ F \mathbf{w} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} (-1 \cdot 3) + (0 \cdot 4) \\ (0 \cdot 3) + (1 \cdot 4) \end{pmatrix} = \begin{pmatrix} -3 \\ 4 \end{pmatrix} \]
O lea la, o le taunuuga o le liua o le vector \( \mathbf{w} \) e le matrix \( F \) o le vector \( \mathbf{w'} = \begin{pmatrix} -3 \\ 4 \end{pmatrix} \).
Fesili 4: Suiga Tuufaatasi
Faapea o loo i ai ni matrices e lua o le suiga, o se matrice o le rotation \( R \) o le tulimanu \( \theta = \frac{\pi}{4} \) ma se matrice o le fua \( S \) e pei ona taua i lalo:
\[ R = \begin{pmatrix} \cos\frac{\pi}{4} & -\sin\frac{\pi}{4} \\ \sin\frac{\pi}{4} & \cos\frac{\pi}{4} \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \]
\[ S = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \]
Tuufaatasi nei suiga ma faaaoga i le vector \( \mathbf{z} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \).
Talanoaga:
Muamua, fuafua le matrix o le suiga tuufaatasi \( RS \):
\[ RS = R \cdot S = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \cdot \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} (\frac{\sqrt{2}}{2} \cdot 2) + (-\frac{\sqrt{2}}{2} \cdot 0) & (\frac{\sqrt{2}}{2} \cdot 0) + (-\frac{\sqrt{2}}{2} \cdot 3) \\ (\frac{\sqrt{2}}{2} \cdot 2) + (\frac{\sqrt{2}}{2} \cdot 0) & (\frac{\sqrt{2}}{2} \cdot 0) + (\frac{\sqrt{2}}{2} \cdot 3) \end{pmatrix} = \begin{pmatrix} \sqrt{2} & -\frac{3\sqrt{2}}{2} \\ \sqrt{2} & \frac{3\sqrt{2}}{2} \end{pmatrix} \]
Ona fa'atele lea o le matrix tu'ufa'atasi \( RS \) i le vector \( \mathbf{z} \):
\[ RS \mathbf{z} = \begin{pmatrix} \sqrt{2} & -\frac{3\sqrt{2}}{2} \\ \sqrt{2} & \frac{3\sqrt{2}}{2} \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} (\sqrt{2} \cdot 1) + (-\frac{3\sqrt{2}}{2} \cdot 1) \\ (\sqrt{2} \cdot 1) + (\frac{3\sqrt{2}}{2} \cdot 1) \end{pmatrix} = \begin{pmatrix} \sqrt{2} – \frac{3\sqrt{2}}{2} \\ \sqrt{2} + \frac{3\sqrt{2}}{2} \end{pmatrix} \]
O lea la, o le taunuuga o le suiga tuufaatasi o le vector \( \mathbf{z} \) e le matrix \( RS \) e faapea:
\[ \mathbf{z'} = \begin{pmatrix} \frac{2\sqrt{2} – 3\sqrt{2}}{2} \\ \sqrt{2} + \frac{3\sqrt{2}}{2} \end{pmatrix} = \begin{pmatrix} -\frac{\sqrt{2}}{2} \\ \frac{5\sqrt{2}}{2} \end{pmatrix} \]
I'uga
I totonu o lenei tusiga, ua matou talanoaina ni faʻataʻitaʻiga o faʻafitauli e faʻaalia ai le faʻaaogaina o matrices mo suiga faʻasolosolo. E taua tele le sao o suiga o matrix i le tele o vaega, aemaise lava ata faʻakomepiuta ma le auʻiliʻiliga o faʻamatalaga. I le malamalama i mea taua o suiga o matrix, e pei o le rotation, scaling, ma le reflection, e mafai ona tatou agai atu i le faʻaaogaina o nei manatu i faʻafitauli e sili atu ona faigata. O le aʻoaʻoina o nei manatu o le a matua aoga lava mo soʻo se tasi o loʻo galue i le matematika, fisiki, poʻo le saienisi komepiuta.
