Ngā Tauira Pātai e Matapaki ana i te Hononga i waenga i ngā Matrices me ngā Whakawhitinga
Pendahuluan
Ko te matihiko he huinga tapawhā rite o ngā tau, ngā huānga rānei kua whakaritea ki ngā rarangi me ngā pou. He whānuitia te whakamahinga o ngā matihiko i roto i ngā momo mara pēnei i te tatauranga, te ahupūngao, te ōhanga, ā, inā koa i roto i ngā panonitanga āhuahanga i roto i te pāngarau me te whakairoiro rorohiko. He taputapu whai hua anō hoki ngā matihiko mō te whakahaere raraunga me te whakaahua me te whakaoti rapanga pāngarau. Ko tētahi whakamahinga nui o ngā matihiko ko ngā panonitanga rārangi, te wāhi e whakamahia ai ngā mahi matihiko hei whakarerekē i te āhua me te tūranga o ngā mea āhuahanga i roto i te wāhi.
I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira rapanga e whakaatu ana i te whakamahinga o ngā matihiko mō ngā panonitanga rārangi, me te whakamārama taipitopito i ngā otinga mō aua rapanga.
Ngā Whakamāramatanga me ngā Tuhituhinga
Hei tīmatanga, me arotake tātou i ētahi whakamāramatanga me ngā tuhipoka taketake ka whakamahia i roto i tēnei kōrero:
1. Matrix: He huinga tau tapawhā rite kua whakaritea ki ngā rarangi me ngā pou.
2. Whakawhitinga Rārangi: He mahi e tango ana i tētahi whārite ka hono atu ki tētahi atu whārite mā te whakamahi i ngā mahi matihiko.
3. Wēkere: He huānga o tētahi huinga wēkere he roa, he ahunga hoki tōna, e whakaaturia ana i te nuinga o te wā hei pou, hei rarangi rānei i roto i tētahi matihiko.
Ko te tuhi i ngā tohu matihiko he mea tuhi ki ngā reta matua, hei tauira \( A \), \( B \), ā, ko ngā vectors he mea tuhi ki ngā reta matotoru, ki ngā pere rānei i runga ake, hei tauira \( \mathbf{v} \) me \( \vec{v} \ rānei).
Ngā Pātai Tauira me te Kōrero
Pātai 1: Hurihanga Hurihanga
I runga i te matatiki whakawhiti hurihuri \( R \) mā te koki \( \theta \) i roto i te wāhi rua-ahu:
\[ R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \]
Wēkere \( \mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \). Whakatauhia te hua o te panonitanga o te wēkere \( \mathbf{v} \) mā te matihiko \( R \) mēnā \( \theta = \frac{\pi}{2} \).
Kōrero:
Tuatahi, whakauruhia ngā uara koki \( \theta = \frac{\pi}{2} \) ki roto i te matihiko \( R \):
\[ R = \begin{pmatrix} \cos\frac{\pi}{2} me -\sin\frac{\pi}{2} \\ \sin\frac{\pi}{2} me \cos\frac{\pi}{2} \end{pmatrix} = \begin{pmatrix} 0 me -1 \\ 1 me 0 \end{pmatrix} \]
Muri iho, whakareatia te matihiko \( R \) ki te ira \( \mathbf{v} \):
\[ R \mathbf{v} = \begin{pmatrix} 0 me te -1 \\ 1 me te 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} (0 \cdot 1) + (-1 \cdot 0) \\ (1 \cdot 1) + (0 \cdot 0) \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]
Nō reira, ko te hua o te whakarerekētanga o te ira \( \mathbf{v} \) e te matihiko \( R \) mō te koki \( \theta = \frac{\pi}{2} \) ko te ira \( \mathbf{v'} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \).
Pātai 2: Te Hurihanga Tauine
I hoatu he matihiko whakawhiti tauine \( S \) i roto i te wāhi rua-ahu penei:
\[ S = \begin{pmatrix} 2 me te 0 \\ 0 me te 3 \end{pmatrix} \]
Wēkere \( \mathbf{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \). Kimihia te hua o te panonitanga o te wēkere \( \mathbf{u} \) mā te matihiko \( S \).
Kōrero:
Whakareatia te matihiko \( S \) ki te ira \( \mathbf{u} \):
\[ S \mathbf{u} = \begin{pmatrix} 2 me te 0 \\ 0 me te 3 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} (2 \cdot 1) + (0 \cdot 2) \\ (0 \cdot 1) + (3 \cdot 2) \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} \]
Nō reira, ko te hua o te whakarerekētanga o te ira \( \mathbf{u} \) e te matihiko \( S \) ko te ira \( \mathbf{u'} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} \).
Pātai 3: Te Hurihanga Whakaaroaro
I runga i te matatiki whakaata \( F \) e pā ana ki te tuaka-y:
\[ F = \begin{pmatrix} -1 me te 0 \\ 0 me te 1 \end{pmatrix} \]
Tātaihia te hua o te whakawhiti i te whārite \( \mathbf{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \) mā te whakamahi i te matihiko whakaata \( F \).
Kōrero:
Whakareatia te matihiko \( F \) ki te ira \( \mathbf{w} \):
\[ F \mathbf{w} = \begin{pmatrix} -1 me te 0 \\ 0 me te 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} (-1 \cdot 3) + (0 \cdot 4) \\ (0 \cdot 3) + (1 \cdot 4) \end{pmatrix} = \begin{pmatrix} -3 \\ 4 \end{pmatrix} \]
Nō reira, ko te hua o te whakarerekētanga o te ira \( \mathbf{w} \) e te matihiko \( F \) ko te ira \( \mathbf{w'} = \begin{pmatrix} -3 \\ 4 \end{pmatrix} \).
Pātai 4: Ngā Huringa Whakakotahi
Me kī e rua ngā matihiko whakawhiti, he matihiko hurihuri \( R \) he koki \( \theta = \frac{\pi}{4} \) me tētahi matihiko tauine \( S \) penei:
\[ R = \begin{pmatrix} \cos\frac{\pi}{4} me te -\sin\frac{\pi}{4} \\ \sin\frac{\pi}{4} me te \cos\frac{\pi}{4} \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} me te -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} me te \frac{\sqrt{2}}{2} \end{pmatrix} \]
\[ S = \begin{pmatrix} 2 me te 0 \\ 0 me te 3 \end{pmatrix} \]
Whakakotahitia ēnei panonitanga ka tāpirihia ki te ira \( \mathbf{z} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \).
Kōrero:
Tuatahi, tatauhia te matihiko whakawhiti whakakotahi \( RS \):
\[ RS = R \cdot S = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \cdot \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} (\frac{\sqrt{2}}{2} \cdot 2) + (-\frac{\sqrt{2}}{2} \cdot 0) & (\frac{\sqrt{2}}{2} \cdot 0) + (-\frac{\sqrt{2}}{2} \cdot 3) \\ (\frac{\sqrt{2}}{2} \cdot 2) + (\frac{\sqrt{2}}{2} \cdot 0) & (\frac{\sqrt{2}}{2} \cdot 0) + (\frac{\sqrt{2}}{2} \cdot 3) \end{pmatrix} = \begin{pmatrix} \sqrt{2} & -\frac{3\sqrt{2}}{2} \\ \sqrt{2} & \frac{3\sqrt{2}}{2} \end{pmatrix} \]
Kātahi, whakareatia te matihiko whakakotahi \( RS \) ki te ira \( \mathbf{z} \):
\[ RS \mathbf{z} = \begin{pmatrix} \sqrt{2} & -\frac{3\sqrt{2}}{2} \\ \sqrt{2} & \frac{3\sqrt{2}}{2} \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} (\sqrt{2} \cdot 1) + (-\frac{3\sqrt{2}}{2} \cdot 1) \\ (\sqrt{2} \cdot 1) + (\frac{3\sqrt{2}}{2} \cdot 1) \end{pmatrix} = \begin{pmatrix} \sqrt{2} – \frac{3\sqrt{2}}{2} \\ \sqrt{2} + \frac{3\sqrt{2}}{2} \end{pmatrix} \]
Nō reira, ko te hua o te panonitanga whakakotahi o te ira \( \mathbf{z} \) e te matihiko \( RS \) ko:
\[ \mathbf{z'} = \begin{pmatrix} \frac{2\sqrt{2} – 3\sqrt{2}}{2} \\ \sqrt{2} + \frac{3\sqrt{2}}{2} \end{pmatrix} = \begin{pmatrix} -\frac{\sqrt{2}}{2} \\ \frac{5\sqrt{2}}{2} \end{pmatrix} \]
Whakamutunga
I roto i tēnei tuhinga, kua matapakihia e mātou ētahi tauira raruraru e whakaatu ana i te whakamahinga o ngā matihiko mō ngā panonitanga rārangi. He mea nui te mahi a ngā panonitanga matihiko i roto i ngā mara maha, inā koa ngā whakairoiro rorohiko me te tātari raraunga. Mā te mārama ki ngā kaupapa matua o ngā panonitanga matihiko, pērā i te hurihanga, te tauine, me te whakaata, ka taea e tātou te neke atu ki te whakamahi i ēnei ariā ki ngā raruraru uaua ake. Ko te matatau ki ēnei ariā he mea tino nui ki te hunga e mahi ana i roto i te pāngarau, te ahupūngao, te pūtaiao rorohiko rānei.
