Zitsanzo za Mafunso Okhudza Ma Vector a Column ndi Ma Vector a Row
Mu masamu, makamaka algebra yolunjika, mavekitala ndi lingaliro lofunikira lomwe limagwiritsidwa ntchito nthawi zambiri m'magwiritsidwe osiyanasiyana, kuyambira pakupanga ma physics mpaka kuwerengera. Mavekitala a m'makola ndi mavekitala a m'makola ndi mitundu iwiri ya mavekitala, iliyonse ili ndi mawonekedwe ake ndi ntchito zake. Nkhaniyi ikambirana za mavuto a zitsanzo ndi mayankho awo okhudzana ndi mavekitala a m'makola ndi mavekitala a m'makola.
Tanthauzo la Vekitala ya Khoma ndi Vekitala ya Mzere
Tisanalowe mu mafunso achitsanzo ndi kukambirana kwawo, choyamba tiyeni tiwone matanthauzidwe oyambira a ma vekitala a m'makola ndi ma vekitala a mzere.
– Ma vekitala a m'mizere ndi ma vekitala okonzedwa mu mzera, kutanthauza kuti, gawo limodzi loyima. Chitsanzo:
\[
\mathbf{v} = \begin{pmatrix}
4 \\
3 \\
2
\end{pmatrix}
\]
– Ma vekitala a mizere ndi ma vekitala okonzedwa m'mizere, kutanthauza kuti, mu gawo limodzi lopingasa. Chitsanzo:
\[
\mathbf{w} = \begin{pmatrix} 5 & 1 & 7 \end{pmatrix}
\]
Chitsanzo 1: Kuwonjezera Ma Vector a Column
Funso:
Popeza ma vector awiri otsatirawa a column:
\[
\mathbf{u} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix}
\]
Werengani chiwerengero cha ma vector awiri a m'magawo awiri.
Yankho:
Kuwonjezera ma vector awiri a m'magawo kumachitika powonjezera zinthu zomwe zikugwirizana nazo.
\[
\mathbf{u} + \mathbf{v} = \kuyamba{pmatrix}
1 \\
2 \\
3
\end{pmatrix} + \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix} = \begin{pmatrix}
1 + 4 \\
2 + 1 \\
3 + 0
\end{pmatrix} = \begin{pmatrix}
5 \\
3 \\
3
\end{pmatrix}
\]
Kotero, chiwerengero cha \(\mathbf{u}\) ndi \(\mathbf{v}\) ndi \(\begin{pmatrix} 5 \\ 3 \\ 3 \end{pmatrix}\).
Chitsanzo Funso 2: Kuwonjezera Ma Vector a Mizere
Funso:
Popeza ma vector awiri otsatirawa a mzere:
\[
\mathbf{a} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 & 3 & 5 \end{pmatrix}
\]
Werengani chiwerengero cha ma vector awiri a mizere.
Yankho:
Kuwonjezera ma vector awiri a mizere kumachitika powonjezera zinthu zofanana.
\[
\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 & 4 + 3 & 6 + 5 \end{pmatrix} = \begin{pmatrix} 3 & 7 & 11 \end{pmatrix}
\]
Kotero, chiwerengero cha \(\mathbf{a}\) ndi \(\mathbf{b}\) ndi \(\begin{pmatrix} 3 & 7 & 11 \end{pmatrix}\).
Chitsanzo 3: Kuchulukitsa kwa Scalar ndi Ma Vector a Column
Funso:
Popeza vekitala ya kolamu \(\mathbf{c}\) ndi scalar \(k\):
\[
\mathbf{c} = \begin{pmatrix}
-3 \\
4 \\
5
\end{pmatrix}, \quad k = 2
\]
Werengerani zotsatira za kuchulukitsa kwa scalar.
Yankho:
Kuchulukitsa kwa scalar ndi vekitala ya column kumachitika pochulukitsa chinthu chilichonse cha vekitala ndi sikitala.
\[
k\mathbf{c} = 2 \kuyamba{pmatrix}
-3 \\
4 \\
5
\end{pmatrix} = \begin{pmatrix}
2 \nthawi -3 \\
Kuwirikiza kawiri 4
2 \nthawi 5
\end{pmatrix} = \begin{pmatrix}
-6 \\
8 \\
10
\end{pmatrix}
\]
Kotero, zotsatira za kuchulukitsa scalar \(2\) ndi vekitala ya column \(\mathbf{c}\) ndi \(\begin{pmatrix} -6 \\ 8 \\ 10 \end{pmatrix}\).
Chitsanzo Funso 4: Kuchulukitsa kwa Scalar ndi Ma Vector a Mizere
Funso:
Kupatsidwa vekitala ya mzere \(\mathbf{d}\) ndi scalar \(m\):
\[
\mathbf{d} = \begin{pmatrix} 7 & -2 & 1 \end{pmatrix}, \quad m = -3
\]
Werengerani zotsatira za kuchulukitsa kwa scalar.
Yankho:
Kuchulukitsa kwa scalar ndi vekitala ya mzere kumachitika pochulukitsa chinthu chilichonse cha vekitala ndi sikitala.
\[
m\mathbf{d} = -3 \begin{pmatrix} 7 & -2 & 1 \end{pmatrix} = \begin{pmatrix} -3 \times 7 & -3 \times -2 & -3 \times 1 \end{pmatrix} = \begin{pmatrix} -21 & 6 & -3 \end{pmatrix}
\]
Kotero, zotsatira za kuchulukitsa scalar \(-3\) ndi row vector \(\mathbf{d}\) ndi \(\begin{pmatrix} -21 & 6 & -3 \end{pmatrix}\).
Chitsanzo 5: Kuchulukitsa kwa Matrix \(1 \nthawi 3\) ndi \(3 \nthawi 1\) (Vekitala ya Mzere ndi Vekitala ya Khoma)
Funso:
Kupatsidwa vekitala ya mzere \(\mathbf{e}\) ndi vekitala ya mzati \(\mathbf{f}\):
\[
\mathbf{e} = \kuyamba{pmatrix} 2 & -1 & 4 \kumapeto{pmatrix}, \quad \mathbf{f} = \kuyamba{pmatrix}
5 \\
3 \\
-2
\end{pmatrix}
\]
Werengerani zotsatira za ma vector awiriwa.
Yankho:
Kuti tichite kuchulukitsa kwa matrix, vekitala ya mzere \(\mathbf{e}\) imawonedwa ngati matrix ya \(1 \times 3\), ndipo vekitala ya m'ndandanda \(\mathbf{f}\) imawonedwa ngati matrix ya \(3 \times 1\). Zotsatira za kuchulukitsa kumeneku ndi scalar, yomwe ndi kuchuluka kwa zinthu zomwe zikugwirizana:
\[
\mathbf{e} \mathbf{f} = \kuyamba{pmatrix} 2 & -1 & 4 \kumapeto{pmatrix} \kuyamba{pmatrix}
5 \\
3 \\
-2
\end{pmatrix} = (2 \nthawi 5) + (-1 \nthawi 3) + (4 \nthawi -2) = 10 – 3 – 8 = -1
\]
Kotero, zotsatira za kuchulukitsa vekitala ya mzere \(\mathbf{e}\) ndi vekitala ya m'ndandanda \(\mathbf{f}\) ndi \(-1\).
Chitsanzo 6: Kuchulukitsa kwa Matrix \(3 \nthawi 1\) ndi \(1 \nthawi 3\) (Column Vector ndi Row Vector)
Funso:
Popatsidwa vekitala ya kolamu \(\mathbf{g}\) ndi vekitala ya mzere \(\mathbf{h}\):
\[
\mathbf{g} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{h} = \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}
\]
Werengerani zotsatira za ma vector awiriwa.
Yankho:
Kuchulukitsa kwa matrix ya vekitala ya column ndi vekitala ya mzere kumapanga matrix ya (\(3 \times 1\)) yochulukitsidwa ndi (\(1 \times 3\)) yomwe imapanga matrix ya \(3 \times 3\). Chinthu chilichonse chatsopano ndi chipatso cha zinthu zake zofanana:
\[
\mathbf{g} \mathbf{h} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix}
1 \nthawi 4 & 1 \nthawi 5 & 1 \nthawi 6 \\
2 \nthawi 4 & 2 \nthawi 5 & 2 \nthawi 6 \\
3 \nthawi 4 & 3 \nthawi 5 & 3 \nthawi 6
\end{pmatrix} = \begin{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12 & 15 & 18
\end{pmatrix}
\]
Kotero, zotsatira za kuchulukitsa vekitala ya kolamu \(\mathbf{g}\) ndi vekitala ya mzere \(\mathbf{h}\) ndi matrix:
\[
\begin{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12 & 15 & 18
\end{pmatrix}
\]
Mapeto
Munkhaniyi yonse, taona zitsanzo zingapo zokhudzana ndi ma vector a m'makola ndi mzere. Kuwonjezera ma vector a m'makola ndi mzere kumachitika powonjezera zinthu zomwe zikugwirizana nazo. Kuchulukitsa kwa scalar ndi vector kumachitikanso pochulukitsa chinthu chilichonse cha vekitala ndi scalar. Pomaliza, taphunzira momwe tingachulukitsire ma vector a m'makola ndi mzere, ndikupanga scalar kapena matrix, kutengera dongosolo lawo. Kudziwa bwino ntchito zoyambira izi ndikofunikira kwambiri pakugwiritsa ntchito zovuta kwambiri mu algebra yolunjika ndi kusanthula deta.
