Tunjukkan bahwa himpunan dari semua matriks berukuran $2 \times 3$ beserta operasi matrix addition dan scalar multiplication merupakan sebuah ruang vektor.
Bukti:
Misalkan himpunan dari semua matriks berukuran $2 \times 3$ beserta operasi matrix addition dan scalar multiplication adalah $V$.
Diketahui juga $A$, $B$, dan $C$ adalah matriks berukuran $2 \times 3$ dan $k$, $l$ adalah skalar dengan spesifikasi sebagai berikut:
\(A = \begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}\), \(B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23}\end{bmatrix}\), dan \(C = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \end{bmatrix}\).
Larson (2016) subbab 4.2 hlm. 161 menyatakan bahwa pembuktian suatu himpunan merupakan ruang vektor harus memenuhi 10 aksioma. Berikut akan dibuktikan untuk 10 aksioma tersebut.
- Apakah $A+B$ juga ada di dalam $V$?
Ya karena
\(\begin{align} A + B &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \\ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \end{bmatrix}. \end{align}\)
\(\begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \\ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \end{bmatrix}\) merupakan matriks berukuran $2 \times 3 $, berarti $A+B$ juga ada di dalam $V$.
- Apakah $A+B = B+A$?
Ya karena
\(\begin{align} A+B &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \\ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \end{bmatrix} \\ &= \begin{bmatrix} b_{11}+a_{11} & b_{12}+a_{12} & b_{13}+a_{13} \\ b_{21}+a_{21} & b_{22}+a_{22} & b_{23}+a_{23} \end{bmatrix} \\ &= \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} + \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= B + A. \end{align}\)
- Apakah $A + (B + C) = (A+B) + C$?
Ya karena
\(\begin{align} A+(B+C) &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \left( \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} + \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \end{bmatrix} \right) \\ &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \left( \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} & b_{13} + c_{13} \\ b_{21} + c_{21} & b_{22} + c_{22} & b_{23} + c_{23} \end{bmatrix} \right) \\ &= \begin{bmatrix} a_{11} + b_{11} + c_{11} & a_{12} + b_{12} + c_{12} & a_{13} + b_{13} + c_{13} \\ a_{21} + b_{21} + c_{21} & a_{22} + b_{22} + c_{22} & a_{23} + b_{23} + c_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \\ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \end{bmatrix} + \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \end{bmatrix} \\ &= \left( \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \right) + \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \end{bmatrix} \\ &= (A+B)+C. \end{align}\)
- Apakah $V$ mempunyai matriks nol $\textbf{0}$ sedemikian sehingga untuk setiap matriks $A$ di $V$, $A + \textbf{0} = A$?
Ada, matriks $\mathbf{0}$ tersebut adalah
\(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\) karena \(A + \mathbf{0} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} = A.\)
- Untuk setiap matriks $A$ di dalam $V$, apakah $V$ mempunyai matriks yang dilambangkan dengan $-A$ sedemikian sehingga $A + (-A) = \mathbf{0}$?
Ada, matriks $-A$ tersebut adalah \(\begin{bmatrix} -a_{11} & -a_{12} & -a_{13} \\ -a_{21} & -a_{22} & -a_{23} \end{bmatrix}\) karena \(\begin{align} A + (-A) &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} -a_{11} & -a_{12} & -a_{13} \\ -a_{21} & -a_{22} & -a_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11}-a_{11} & a_{12}-a_{12} & a_{13}-a_{13} \\ a_{21}-a_{21} & a_{22}-a_{22} & a_{23}-a_{23} \end{bmatrix} \\ &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ &= \mathbf{0}. \end{align}\)
- Apakah $kA$ juga ada di dalam $V$?
Ya, karena \(kA = k \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} = \begin{bmatrix} ka_{11} & ka_{12} & ka_{13} \\ ka_{21} & ka_{22} & ka_{23} \end{bmatrix}.\)
\(\begin{bmatrix} ka_{11} & ka_{12} & ka_{13} \\ ka_{21} & ka_{22} & ka_{23} \end{bmatrix}\) adalah matriks berukuran $2 \times 3$ sehingga $cA$ juga berada di dalam $V$.
- Apakah $k(A+B) = kA + kB$?
Ya karena
\(\begin{align} k(A+B) &= k \left( \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \right) \\ &= k \left( \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \\ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \end{bmatrix} \right) \\ &= \begin{bmatrix} k(a_{11} + b_{11}) & k(a_{12} + b_{12}) & k(a_{13} + b_{13}) \\ k(a_{21} + b_{21}) & k(a_{22} + b_{22}) & k(a_{23} + b_{23}) \end{bmatrix} \\ &= \begin{bmatrix} ka_{11} + kb_{11} & ka_{12} + kb_{12} & ka_{13} + kb_{13} \\ ka_{21} + kb_{21} & ka_{22} + kb_{22} & ka_{23} + kb_{23} \end{bmatrix} \\ &= \begin{bmatrix} ka_{11} & ka_{12} & ka_{13} \\ ka_{21} & ka_{22} & ka_{23} \end{bmatrix} + \begin{bmatrix} kb_{11} & kb_{12} & kb_{13} \\ kb_{21} & kb_{22} & kb_{23} \end{bmatrix} \\ &= k \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + k \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \\ &= kA + kB. \end{align}\)
- Apakah $(k+l)A = kA + lA$?
Ya karena
\(\begin{align} (k+l) A &= (k+l) \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ \end{bmatrix} \\ &= \begin{bmatrix} (k+l) a_{11} & (k+l) a_{12} & (k+l) a_{13} \\ (k+l) a_{21} & (k+l) a_{22} & (k+l) a_{23} \end{bmatrix} \\ &= \begin{bmatrix} k a_{11} + l a_{11} & k a_{12} + l a_{12} & k a_{13} + l a_{13} \\ k a_{21} + l a_{21} & k a_{22} + l a_{22} & k a_{23} + l a_{23} \end{bmatrix} \\ &= \begin{bmatrix} k a_{11} & k a_{12} & k a_{13} \\ k a_{21} & k a_{22} & k a_{23} \end{bmatrix} + \begin{bmatrix} l a_{11} & l a_{12} & l a_{13} \\ l a_{21} & l a_{22} & l a_{23} \end{bmatrix} \\ &= k \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + l \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= kA + lA. \end{align}\)
- Apakah $k(lA) = (kl)A$?
Ya karena
\(\begin{align} k(l A) &= k \left( l \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \right) \\ &= k \left( \begin{bmatrix} l a_{11} & l a_{12} & l a_{13} \\ l a_{21} & l a_{22} & l a_{23} \end{bmatrix} \right) \\ &= k \begin{bmatrix} l a_{11} & l a_{12} & l a_{13} \\ l a_{21} & l a_{22} & l a_{23} \end{bmatrix} \\ &= \begin{bmatrix} kl a_{11} & kl a_{12} & kl a_{13} \\ kl a_{21} & kl a_{22} & kl a_{23} \end{bmatrix} \\ &= \begin{bmatrix} (kl)a_{11} & (kl)a_{12} & (kl)a_{13} \\ (kl)a_{21} & (kl)a_{22} & (kl)a_{23} \end{bmatrix} \\ &= (kl) \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= (kl) A. \end{align}\)
- Apakah $1(A) = A$?
Ya karena
\(\begin{align} 1(A) &= 1 \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= \begin{bmatrix} 1a_{11} & 1a_{12} & 1a_{13} \\ 1a_{21} & 1a_{22} & 1a_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= A. \end{align}\)
Karena $V$ memenuhi 10 aksioma dari definisi suatu ruang vektor, $V$ atau himpunan dari semua matriks berukuran $2 \times 3$ beserta operasi matrix addition dan scalar multiplication merupakan sebuah ruang vektor. $\square$