Ngā Tauira Pātai mō te Matapaki i te Hanganga Whakawhitiwhiti mā te Whakamahi i ngā Matrices
He kaupapa nui ngā panonitanga āhuahanga i roto i te pāngarau, inā koa i roto i te āhuahanga me te ararau rārangi. Ka uru atu pea ēnei panonitanga ki ngā whakawhiti, ngā hurihanga, ngā whakaata, me ngā whakawhanuitanga. I roto i tēnei tuhinga, ka tirohia e mātou me pēhea te whakaatu me te whakaoti i te hanganga o ngā panonitanga mā te whakamahi i ngā matihiko. Ka whakaratohia hoki e mātou he tauira rapanga me ngā otinga.
1. Whakataki ki te Whakawhitiwhitinga mā te whakamahi i ngā Matrices
Ka taea te whakaatu i ngā panonitanga āhuahanga mā roto i ngā matihiko. Hei tauira, ka taea te whakatakoto i ngā panonitanga hurihanga, whakawhiti, whakaata, me te whakawhānui ki te āhua matihiko penei:
1. Whakamāoritanga
\[
T(x, y) = \begin{pmatrix} x + a \\ y + b \end{pmatrix}
\]
2. Hurihanga
\[
R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}
\]
3. Te whakaata i te tuaka-X
\[
\text{Whakaata X} = \begin{pmatrix} 1 me te 0 \\ 0 me te -1 \end{pmatrix}
\]
4. Whakawhanui (whakanui/whakanui)
\[
D(k) = \begin{pmatrix} k me te 0 \\ 0 me te k \end{pmatrix}
\]
2. Te Hanganga o ngā Huringa me ngā Matrices
Ko te tito whakawhiti ko te whakamahinga raupapa o ngā whakawhiti e rua, neke atu rānei, ki tētahi mea. Hei tatau i te tito whakawhiti mā te whakamahi i ngā matihiko, ka whakareatia noa e mātou ngā matihiko e tohu ana i ngā whakawhiti.
Ngā Pātai Tauira me te Kōrero
Pātai
I te pūwāhi P(2, 3), kimihia te hua o te panonitanga e whai ake nei:
1. Hurihanga 90^\circ\) ki te taha matau (CW)
2. Whakawhanui me te tauine tauine o te 2
3. Te whakamāoritanga o (1, -2)
Kōrero
1. Hurihanga \(90^\circ\) CW
Ko te matihiko mō te hurihanga matau o \(90^\circ\):
\[
\begin{pmatrix} \cos(-90^\circ) me te -\sin(-90^\circ) \\ \sin(-90^\circ) me te \cos(-90^\circ) \end{pmatrix} = \begin{pmatrix} 0 me te 1 \\ -1 me te 0 \end{pmatrix}
\]
Te whakamahi i te panonitanga hurihuri ki te pūwāhi P:
\[
\begin{pmatrix} 0 me te 1 \\ -1 me te 0 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 0 \cdot 2 + 1 \cdot 3 \\ -1 \cdot 2 + 0 \cdot 3 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}
\]
Ko te pūwāhi P i muri i te panonitanga hurihuri ko P'(3, -2).
2. Whakawhanui me te tauine tauine o te 2
Te matihiko mō te whakawhanuitanga me te tauine tauine 2:
\[
\begin{pmatrix} 2 me te 0 \\ 0 me te 2 \end{pmatrix}
\]
Te whakamahi i te whakawhiti whānui i te pūwāhi P'(3, -2):
\[
\begin{pmatrix} 2 me te 0 \\ 0 me te 2 \end{pmatrix} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 2 \cdot 3 + 0 \cdot -2 \\ 0 \cdot 3 + 2 \cdot -2 \end{pmatrix} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}
\]
Ko te pūwāhi P' i muri i te whakawhiti whānui ko P”(6, -4).
3. Te whakamāoritanga o (1, -2)
Anei ngā mahi whakamāori kua hoatu:
\[
T(x, y) = \begin{pmatrix} x + 1 \\ y – 2 \end{pmatrix}
\]
Te whakamahi i te whakawhiti whakawhiti i te pūwāhi P”(6, -4):
\[
T(6, -4) = \begin{pmatrix} 6 + 1 \\ -4 – 2 \end{pmatrix} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}
\]
Nō reira, ko te pūwāhi mutunga i muri i te whakamahinga o ngā panonitanga katoa ko P(7, -6).
3. Te Tātai i te Hanganga Whakawhiti
Ngā Pātai Tāpiri
Homai te pūwāhi Q(1, 2) me te panonitanga e whai ake nei:
1. Te whakaaroaro mō te tuaka-X.
2. Hurihanga \(180^\circ\) ki te taha matau (CW).
Kōrero
1. Te whakaata i te tuaka-X
Te matihiko whakaata mō te tuaka-X:
\[
\begin{pmatrix} 1 me te 0 \\ 0 me te -1 \end{pmatrix}
\]
Te whakamahi i tētahi panoni whakaata i te pūwāhi Q:
\[
\begin{pmatrix} 1 me te 0 \\ 0 me te -1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 0 \cdot 2 \\ 0 \cdot 1 + (-1) \cdot 2 \end{pmatrix} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}
\]
Ko te pūwāhi Q i muri i te panonitanga whakaata ko Q'(1, -2).
2. Hurihanga \(180^\circ\) CW
Matrix mō te hurihanga \(180^\circ\) i te taha matau:
\[
\begin{pmatrix} \cos(180^\circ) me te -\sin(180^\circ) \\ \sin(180^\circ) me te \cos(180^\circ) \end{pmatrix} = \begin{pmatrix} -1 me te 0 \\ 0 me te -1 \end{pmatrix}
\]
Te whakamahi i te panoni hurihuri \(180^\circ\) ki te pūwāhi Q'(1, -2):
\[
\begin{pmatrix} -1 me te 0 \\ 0 me te -1 \end{pmatrix} \begin{pmatrix} 1 \\ -2 \end{pmatrix} = \begin{pmatrix} -1 \cdot 1 + 0 \cdot -2 \\ 0 \cdot 1 + -1 \cdot -2 \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}
\]
Nō reira, ko te pūwāhi mutunga i muri i te whakamahinga o ngā panonitanga katoa ko Q(-1, 2).
Te Katinga
He tino whai hua te tikanga tito whakawhiti mā te whakamahi i ngā matihiko hei whakangawari me te tatau pūnaha i ngā whakawhiti āhuahanga. Mā te whai i ngā kaupae i runga ake nei, ka taea e tātou te mārama me te whakamahi i ngā momo whakawhiti ki tētahi pūwāhi kotahi, ki tētahi atu mea āhuahanga rānei. Mā te ako ki te whakamahi i ngā matihiko i roto i ngā whakawhiti ka māmā ake te whakamahi i aua whakawhiti ki ngā momo mara pērā i te ahupūngao, te whakairoiro rorohiko, me ētahi atu.
