Fa'ata'ita'iga o Fesili e Talanoaina ai Vectors o Koluma ma Vectors o Laina
I le matematika, aemaise lava le algebra linear, o vectors o se manatu faavae e masani ona faʻaaogaina i le tele o faʻaoga, mai le faʻataʻitaʻiga o le fisiki i le faʻatusatusaga. O vectors koluma ma vectors laina o ni ituaiga se lua o le faʻatusaina o vector, e tofu ma ona uiga ma faʻaoga. O lenei tusiga o le a talanoaina ai faʻataʻitaʻiga o faʻafitauli ma a latou fofo e aofia ai vectors koluma ma vectors laina.
Fa'amatalaga o le Column Vector ma le Row Vector
A o le’i oo atu i fesili fa’ata’ita’i ma a latou talanoaga, se’i o tatou toe iloilo muamua fa’amatalaga faavae o vectors o koluma ma vectors o laina.
– O vectors o koluma o vectors ia o loʻo faʻatulagaina i totonu o se koluma, o lona uiga, o le tasi le fua faʻatulagaina. Faʻataʻitaʻiga:
\[
\mathbf{v} = \begin{pmatrix}
4
3
2
\end{pmatrix}
\]
– O vectors o laina o vectors ia o loʻo faʻatulagaina i laina, o lona uiga, i le tasi le lapoʻa faʻalava. Faʻataʻitaʻiga:
\[
\mathbf{w} = \begin{pmatrix} 5 & 1 & 7 \end{pmatrix}
\]
Faataitaiga 1: Faaopoopoina o Vectors o le Koluma
Fesili:
I le tuuina atu o vectors koluma e lua nei:
\[
\mathbf{u} = \begin{pmatrix}
1
2
3
\end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix}
4
1
0
\end{pmatrix}
\]
Fuafua le aofaiga o vectors koluma e lua.
Tali:
O le faʻaopoopoga o vectors koluma e lua e faia e ala i le faʻaopoopoina o a la elemene talafeagai.
\[
\mathbf{u} + \mathbf{v} = \amata{pmatrix}
1
2
3
\end{pmatrix} + \begin{pmatrix}
4
1
0
\end{pmatrix} = \begin{pmatrix}
1 + 4
2 + 1
3 + 0
\end{pmatrix} = \begin{pmatrix}
5
3
3
\end{pmatrix}
\]
O lea la, o le aofaʻi o \(\mathbf{u}\) ma le \(\mathbf{v}\) o le \(\begin{pmatrix} 5 \\ 3 \\ 3 \end{pmatrix}\).
Fa'ata'ita'iga Fesili 2: Fa'aopoopoina o Vectors o Laina
Fesili:
I le tuuina atu o laina e lua o loo mulimuli mai:
\[
\mathbf{a} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 & 3 & 5 \end{pmatrix}
\]
Fuafua le aofa'i o laina vectors e lua.
Tali:
O le faʻaopoopoga o laina vectors e lua e faia e ala i le faʻaopoopoina o elemene talafeagai.
\[
\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 & 4 + 3 & 6 + 5 \end{pmatrix} = \begin{pmatrix} 3 & 7 & 11 \end{pmatrix}
\]
O lea la, o le aofaʻi o \(\mathbf{a}\) ma le \(\mathbf{b}\) o le \(\begin{pmatrix} 3 & 7 & 11 \end{pmatrix}\).
Faʻataʻitaʻiga 3: Faʻatelega Scalar e ala i Vectors Koluma
Fesili:
I le tuuina atu o se vector koluma \(\mathbf{c}\) ma se scalar \(k\):
\[
\mathbf{c} = \begin{pmatrix}
-3
4
5
\end{pmatrix}, \quad k = 2
\]
Fuafua le iʻuga o le faʻatelega scalar.
Tali:
O le fa'atelega scalar i se column vector e faia e ala i le fa'atelega o elemene ta'itasi o le vector i se scalar.
\[
k\mathbf{c} = 2 \amata{pmatrix}
-3
4
5
\end{pmatrix} = \begin{pmatrix}
2 \times -3 \\
2 taimi 4
2 \ taimi 5
\end{pmatrix} = \begin{pmatrix}
-6
8
10
\end{pmatrix}
\]
O lea la, o le taunuuga o le faateleina o le scalar \(2\) i le column vector \(\mathbf{c}\) o le \(\begin{pmatrix} -6 \\ 8 \\ 10 \end{pmatrix}\).
Fa'ata'ita'iga Fesili 4: Fa'atelega Scalar e ala i Vectors o Laina
Fesili:
I le tuuina atu o se laina vector \(\mathbf{d}\) ma se scalar \(m\):
\[
\mathbf{d} = \begin{pmatrix} 7 & -2 & 1 \end{pmatrix}, \quad m = -3
\]
Fuafua le iʻuga o le faʻatelega scalar.
Tali:
O le fa'atelega scalar i se laina vector e faia e ala i le fa'atelega o elemene ta'itasi o le vector i se scalar.
\[
m\mathbf{d} = -3 \begin{pmatrix} 7 & -2 & 1 \end{pmatrix} = \begin{pmatrix} -3 \times 7 & -3 \times -2 & -3 \times 1 \end{pmatrix} = \begin{pmatrix} -21 & 6 & -3 \end{pmatrix}
\]
O lea la, o le taunuuga o le faateleina o le scalar \(-3\) i le row vector \(\mathbf{d}\) o le \(\begin{pmatrix} -21 & 6 & -3 \end{pmatrix}\).
Fa'ata'ita'iga 5: Fa'atelega o le Matrix \(1 \times 3\) i le \(3 \times 1\) (Vector o le Laina i le Vector o le Koluma)
Fesili:
Tuuina atu se laina vete \(\mathbf{e}\) ma se koluma vete \(\mathbf{f}\):
\[
\mathbf{e} = \begin{pmatrix} 2 & -1 & 4 \end{pmatrix}, \quad \mathbf{f} = \begin{pmatrix}
5
3
-2
\end{pmatrix}
\]
Fuafua le fua fa'atatau o vectors e lua.
Tali:
Mo le fa'atinoina o le fa'atelega o le matrix, o le row vector \(\mathbf{e}\) e fa'atatauina o se \(1 \times 3\) matrix, ma o le column vector \(\mathbf{f}\) e fa'atatauina o se \(3 \times 1\) matrix. O le i'uga o lenei fa'atelega o se scalar, o lona uiga o le aofa'i o oloa o elemene talafeagai:
\[
\mathbf{e} \mathbf{f} = \begin{pmatrix} 2 & -1 & 4 \end{pmatrix} \begin{pmatrix}
5
3
-2
\end{pmatrix} = (2 \times 5) + (-1 \times 3) + (4 \times -2) = 10 – 3 – 8 = -1
\]
O lea la, o le taunuuga o le faateleina o le vector laina \(\mathbf{e}\) i le vector koluma \(\mathbf{f}\) o le \(-1\).
Fa'ata'ita'iga 6: Fa'atelega o le Matrix \(3 \times 1\) i le \(1 \times 3\) (Vector o le Koluma e le Vector o le Laina)
Fesili:
I le tuuina atu o se vector koluma \(\mathbf{g}\) ma se vector laina \(\mathbf{h}\):
\[
\mathbf{g} = \begin{pmatrix}
1
2
3
\end{pmatrix}, \quad \mathbf{h} = \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}
\]
Fuafua le fua fa'atatau o vectors e lua.
Tali:
O le fa'ateleina o le matrix o se column vector i se row vector e maua ai se (\(3 \times 1\)) matrix fa'ateleina i le (\(1 \times 3\)) lea e maua ai se \(3 \times 3\) matrix. O elemene fou ta'itasi o le oloa lea o ona elemene talafeagai:
\[
\mathbf{g} \mathbf{h} = \begin{pmatrix}
1
2
3
\end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix}
1 \ taimi 4 & 1 \ taimi 5 & 1 \ taimi 6 \\
2 \ taimi 4 & 2 \ taimi 5 & 2 \ taimi 6 \\
3 \ taimi 4 & 3 \ taimi 5 & 3 \ taimi 6
\end{pmatrix} = \begin{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12&15&18
\end{pmatrix}
\]
O lea la, o le taunuuga o le faateleina o le column vector \(\mathbf{g}\) i le row vector \(\mathbf{h}\) o le matrix lea:
\[
\begin{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12&15&18
\end{pmatrix}
\]
I'uga
I totonu o lenei tusiga, ua tatou vaʻaia ni faʻataʻitaʻiga se tele e aofia ai vectors o le koluma ma le laina. O le faʻaopoopoga o vectors o le koluma ma le laina e ausia e ala i le faʻaopoopoina o a la elemene talafeagai. O le faʻateleina o le scalar e se vector e ausia foʻi e ala i le faʻateleina o elemene taʻitasi o le vector i le scalar. Ma le mea mulimuli, ua tatou aʻoaʻoina le auala e faʻateleina ai vectors o le laina ma le koluma, ma maua ai se scalar poʻo se matrix, e faʻatatau i la latou faʻasologa. O le puleaina o nei galuega faʻavae e taua tele mo faʻaoga sili atu ona faigata i le linear algebra ma le auʻiliʻiliga o faʻamatalaga.
