Ngā Tauira Pātai e Matapaki ana i ngā Wētera Pou me ngā Wētera Rārangi
I roto i te pāngarau, inā koa te matawai rārangi, he ariā taketake ngā whārite e whakamahia whānuitia ana i roto i ngā tono maha, mai i te whakatauira ahupūngao ki te tatau. Ko ngā whārite pou me ngā whārite rarangi he momo whakaaturanga whārite e rua, he āhuatanga me ngā whakamahinga motuhake tō ia. Ka matapakihia e tēnei tuhinga ngā tauira rapanga me ā rātou otinga e pā ana ki ngā whārite pou me ngā whārite rarangi.
Te Whakamāramatanga o te Wētera Pou me te Wētera Rārangi
I mua i te urunga atu ki ngā tauira pātai me tā rātou matapakinga, me arotake tuatahi tātou i ngā whakamāramatanga taketake o ngā whārite pou me ngā whārite rarangi.
– Ko ngā whārite pou he whārite kua whakaritea ki roto i tētahi pou, arā, kotahi te taha poutū. Hei tauira:
\[
\mathbf{v} = \begin{pmatrix}
4
3
2
\end{pmatrix}
\]
– Ko ngā whārite rarangi he whārite kua whakaritea ki ngā rarangi, arā, i roto i te taha whakapae kotahi. Hei tauira:
\[
\mathbf{w} = \begin{pmatrix} 5 me te 1 me te 7 \end{pmatrix}
\]
Tauira 1: Te Tāpiri i ngā Wētera Pou
Pātai:
I runga i ngā whārite pou e rua e whai ake nei:
\[
\mathbf{u} = \begin{pmatrix}
1
2
3
\end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix}
4
1
0
\end{pmatrix}
\]
Tātaihia te tapeke o ngā whārite pou e rua.
Otinga:
Ka taea te tāpiri i ngā wekere pou e rua mā te tāpiri i ngā huānga e rite ana ki ō rāua.
\[
\mathbf{u} + \mathbf{v} = \begin{pmatrix}
1
2
3
\end{pmatrix} + \begin{pmatrix}
4
1
0
\end{pmatrix} = \begin{pmatrix}
1 + 4
2 + 1
3 + 0
\end{pmatrix} = \begin{pmatrix}
5
3
3
\end{pmatrix}
\]
Nō reira, ko te tapeke o \(\mathbf{u}\) me \(\mathbf{v}\) ko \(\begin{pmatrix} 5 \\ 3 \\ 3 \end{pmatrix}\).
Tauira Pātai 2: Te Tāpiri i ngā Rārangi Wetere
Pātai:
I runga i ngā rarangi e rua e whai ake nei:
\[
\mathbf{a} = \begin{pmatrix} 2 me te 4 me te 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 me te 3 me te 5 \end{pmatrix}
\]
Tātaihia te tapeke o ngā rarangi whārite e rua.
Otinga:
Ka taea te tāpiri i ngā whārite rarangi e rua mā te tāpiri i ngā huānga e rite ana.
\[
\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 & 4 + 3 & 6 + 5 \end{pmatrix} = \begin{pmatrix} 3 & 7 & 11 \end{pmatrix}
\]
Nō reira, ko te tapeke o \(\mathbf{a}\) me \(\mathbf{b}\) ko \(\begin{pmatrix} 3 me te 7 me te 11 \end{pmatrix}\).
Tauira 3: Te Whakarea Tauine mā ngā Wētera Pou
Pātai:
Mēnā ka hoatu he whārite pou \(\mathbf{c}\) me te tauine \(k\):
\[
\mathbf{c} = \begin{pmatrix}
-3
4
5
\end{pmatrix}, \quad k = 2
\]
Tātaihia te hua o te whakareatanga tauine.
Otinga:
Ko te whakarea tauine mā te whakarea i ia huānga o te tauine ki te tauine.
\[
k\mathbf{c} = 2 \begin{pmatrix}
-3
4
5
\end{pmatrix} = \begin{pmatrix}
2 \ngā -3 \\
2 ngā wā 4
2 \ whakareatia ki te 5
\end{pmatrix} = \begin{pmatrix}
-6
8
10
\end{pmatrix}
\]
Nō reira, ko te hua o te whakarea i te tauine \(2\) ki te whārite pou \(\mathbf{c}\) ko \(\begin{pmatrix} -6 \\ 8 \\ 10 \end{pmatrix}\).
Tauira Pātai 4: Te Whakarea Tauine mā ngā Waehere Rārangi
Pātai:
Mēnā ka hoatu he rarangi whārite \(\mathbf{d}\) me te tauine \(m\):
\[
\mathbf{d} = \begin{pmatrix} 7 me -2 me 1 \end{pmatrix}, \quad m = -3
\]
Tātaihia te hua o te whakareatanga tauine.
Otinga:
Ko te whakarea tauine mā te whakarea i ia huānga o te tauine ki te tauine.
\[
m\mathbf{d} = -3 \begin{pmatrix} 7 me te -2 me te 1 \end{pmatrix} = \begin{pmatrix} -3 \times 7 me te -3 \times -2 me te -3 \times 1 \end{pmatrix} = \begin{pmatrix} -21 me te 6 me te -3 \end{pmatrix}
\]
Nō reira, ko te hua o te whakarea i te tauine \(-3\) ki te whārite rarangi \(\mathbf{d}\) ko \(\begin{pmatrix} -21 me 6 me -3 \end{pmatrix}\).
Tauira 5: Te Whakareatanga o te Matrix \(1 \times 3\) ki te \(3 \times 1\) (Te Vector Rārangi mā te Vector Pou)
Pātai:
Hoatu he vector haupae \(\mathbf{e}\) me te vector tīwae \(\mathbf{f}\):
\[
\mathbf{e} = \begin{pmatrix} 2 & -1 & 4 \end{pmatrix}, \quad \mathbf{f} = \begin{pmatrix}
5
3
-2
\end{pmatrix}
\]
Tātaihia te hua o ngā whārite e rua.
Otinga:
Hei whakahaere i te whakarea matihiko, ka kiia te ira rarangi \(\mathbf{e}\) he matihiko \(1 \times 3\), ā, ka kiia te ira pou \(\mathbf{f}\) he matihiko \(3 \times 1\). Ko te hua o tēnei whakarea he tauine, arā, ko te tapeke o ngā hua o ngā huānga e rite ana:
\[
\mathbf{e} \mathbf{f} = \begin{pmatrix} 2 & -1 & 4 \end{pmatrix} \begin{pmatrix}
5
3
-2
\end{pmatrix} = (2 \times 5) + (-1 \times 3) + (4 \times -2) = 10 – 3 – 8 = -1
\]
Nō reira, ko te hua o te whakarea i te whārite rarangi \(\mathbf{e}\) ki te whārite pou \(\mathbf{f}\) ko \(-1\).
Tauira 6: Te Whakareatanga o te Matrix \(3 \times 1\) ki \(1 \times 3\) (Te Wāhanga Pou mā te Wāhanga Rārangi)
Pātai:
I runga i te whārite pou \(\mathbf{g}\) me te whārite rarangi \(\mathbf{h}\):
\[
\mathbf{g} = \begin{pmatrix}
1
2
3
\end{pmatrix}, \quad \mathbf{h} = \begin{pmatrix} 4 me te 5 me te 6 \end{pmatrix}
\]
Tātaihia te hua o ngā whārite e rua.
Otinga:
Mā te whakareatanga matihiko o tētahi whārite pou ki tētahi whārite rarangi ka puta he matihiko (\(3 \times 1\)) ka whakareatia ki (\(1 \times 3\)) ka puta he matihiko \(3 \times 3\). Ko ia huānga hou te hua o ōna huānga e rite ana:
\[
\mathbf{g} \mathbf{h} = \begin{pmatrix}
1
2
3
\end{pmatrix} \begin{pmatrix} 4 me te 5 me te 6 \end{pmatrix} = \begin{pmatrix}
1 \ngā 4 me 1 \ngā 5 me 1 \ngā 6 \ngā
2 \ngā 4 me 2 \ngā 5 me 2 \ngā 6 \ngā
3 \ ngā wā 4 me te 3 \ ngā wā 5 me te 3 \ ngā wā 6
\end{pmatrix} = \begin{pmatrix}
4 me te 5 me te 6
8 me te 10 me te 12
12 & 15 & 18
\end{pmatrix}
\]
Nō reira, ko te hua o te whakarea i te whārite pou \(\mathbf{g}\) ki te whārite rarangi \(\mathbf{h}\) ko te matihiko:
\[
\begin{pmatrix}
4 me te 5 me te 6
8 me te 10 me te 12
12 & 15 & 18
\end{pmatrix}
\]
Whakamutunga
I roto i tēnei tuhinga, kua kite tātou i ētahi tauira e pā ana ki ngā whārite pou me ngā whārite rarangi. Ka taea te tāpiri i ngā whārite pou me ngā whārite rarangi mā te tāpiri i ō rāua huānga e rite ana. Ka taea hoki te whakarea tauine mā te whārite mā te whakarea i ia huānga o te whārite mā te tauine. Hei whakamutunga, kua ako tātou me pēhea te whakarea i ngā whārite rarangi me ngā whārite pou, ka puta he tauine, he matihiko rānei, i runga i tō rātou raupapa. He mea nui te matatau ki ēnei mahi taketake mō ngā tono uaua ake i roto i te arorangi rārangi me te tātari raraunga.
