Hendra Bunyamin

Hendra Bunyamin

him

A Forgiven Sinner, an Associate Professor

Contoh Pembuktian Ruang Vektor dengan sangat Detil

8 minute read

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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.

  1. 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 \(V\).

  2. Apakah \(A+B = B+A\)?
    Ya karena
    \(\begin{align} A+B &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \\ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \end{bmatrix} \\ &= \begin{bmatrix} b_{11}+a_{11} & b_{12}+a_{12} & b_{13}+a_{13} \\ b_{21}+a_{21} & b_{22}+a_{22} & b_{23}+a_{23} \end{bmatrix} \\ &= \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} + \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= B + A. \end{align}\)

  3. Apakah \(A + (B + C) = (A+B) + C\)?
    Ya karena
    \(\begin{align} A+(B+C) &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \left( \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} + \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \end{bmatrix} \right) \\ &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \left( \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} & b_{13} + c_{13} \\ b_{21} + c_{21} & b_{22} + c_{22} & b_{23} + c_{23} \end{bmatrix} \right) \\ &= \begin{bmatrix} a_{11} + b_{11} + c_{11} & a_{12} + b_{12} + c_{12} & a_{13} + b_{13} + c_{13} \\ a_{21} + b_{21} + c_{21} & a_{22} + b_{22} + c_{22} & a_{23} + b_{23} + c_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \\ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \end{bmatrix} + \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \end{bmatrix} \\ &= \left( \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \right) + \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \end{bmatrix} \\ &= (A+B)+C. \end{align}\)

  4. 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 \(A + \mathbf{0} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} = A.\)

  5. 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 \(\begin{align} A + (-A) &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} -a_{11} & -a_{12} & -a_{13} \\ -a_{21} & -a_{22} & -a_{23} \end{bmatrix} \\ &= \begin{bmatrix} a_{11}-a_{11} & a_{12}-a_{12} & a_{13}-a_{13} \\ a_{21}-a_{21} & a_{22}-a_{22} & a_{23}-a_{23} \end{bmatrix} \\ &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ &= \mathbf{0}. \end{align}\)

  6. 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 \(V\).

  7. Apakah \(k(A+B) = kA + kB\)?
    Ya karena
    \(\begin{align} k(A+B) &= k \left( \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \right) \\ &= k \left( \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \\ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \end{bmatrix} \right) \\ &= \begin{bmatrix} k(a_{11} + b_{11}) & k(a_{12} + b_{12}) & k(a_{13} + b_{13}) \\ k(a_{21} + b_{21}) & k(a_{22} + b_{22}) & k(a_{23} + b_{23}) \end{bmatrix} \\ &= \begin{bmatrix} ka_{11} + kb_{11} & ka_{12} + kb_{12} & ka_{13} + kb_{13} \\ ka_{21} + kb_{21} & ka_{22} + kb_{22} & ka_{23} + kb_{23} \end{bmatrix} \\ &= \begin{bmatrix} ka_{11} & ka_{12} & ka_{13} \\ ka_{21} & ka_{22} & ka_{23} \end{bmatrix} + \begin{bmatrix} kb_{11} & kb_{12} & kb_{13} \\ kb_{21} & kb_{22} & kb_{23} \end{bmatrix} \\ &= k \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + k \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \\ &= kA + kB. \end{align}\)

  8. Apakah \((k+l)A = kA + lA\)?
    Ya karena
    \(\begin{align} (k+l) A &= (k+l) \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ \end{bmatrix} \\ &= \begin{bmatrix} (k+l) a_{11} & (k+l) a_{12} & (k+l) a_{13} \\ (k+l) a_{21} & (k+l) a_{22} & (k+l) a_{23} \end{bmatrix} \\ &= \begin{bmatrix} k a_{11} + l a_{11} & k a_{12} + l a_{12} & k a_{13} + l a_{13} \\ k a_{21} + l a_{21} & k a_{22} + l a_{22} & k a_{23} + l a_{23} \end{bmatrix} \\ &= \begin{bmatrix} k a_{11} & k a_{12} & k a_{13} \\ k a_{21} & k a_{22} & k a_{23} \end{bmatrix} + \begin{bmatrix} l a_{11} & l a_{12} & l a_{13} \\ l a_{21} & l a_{22} & l a_{23} \end{bmatrix} \\ &= k \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} + l \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= kA + lA. \end{align}\)

  9. Apakah \(k(lA) = (kl)A\)?
    Ya karena
    \(\begin{align} k(l A) &= k \left( l \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \right) \\ &= k \left( \begin{bmatrix} l a_{11} & l a_{12} & l a_{13} \\ l a_{21} & l a_{22} & l a_{23} \end{bmatrix} \right) \\ &= k \begin{bmatrix} l a_{11} & l a_{12} & l a_{13} \\ l a_{21} & l a_{22} & l a_{23} \end{bmatrix} \\ &= \begin{bmatrix} kl a_{11} & kl a_{12} & kl a_{13} \\ kl a_{21} & kl a_{22} & kl a_{23} \end{bmatrix} \\ &= \begin{bmatrix} (kl)a_{11} & (kl)a_{12} & (kl)a_{13} \\ (kl)a_{21} & (kl)a_{22} & (kl)a_{23} \end{bmatrix} \\ &= (kl) \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \\ &= (kl) A. \end{align}\)

  10. 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. \(\square\)

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