Perubahan Basis dan Pembahasan

 

Perubahan Basis (blogaritma.net)

Secara Umum:Perubahan Basis di R²
Misal S dan S’ Basis di R² dimana S = { u₁,u₂} dan S’ = { v₁,v₂}
i) Perhatikan basis S = { u₁,u₂}
    Vektor di R² yang direntang oleh S = w₁,v₁ dan v₂
Misalkan  :   w = k₁u₁ + k₂ u₂ ; k₁ dan k₂ skalar
                    v₁= a₁u₁ + a₂ u₂ ; a₁ dan a₂ skalar
                    v₂= b₁u₁ + b₂ u₂ ; b₁ dan b₂ skalar
ii) Perhatikan Basis S’ = { v₁,v₂}
    Vektor di R² yang direntang oleh S’ = w₁ , u₁ dan u₂
Misalkan  : w = L₁v₁ + L₂v₂ ; L₁ dan L₂ skalar
                    u₁= c₁v₁ + c₂v₂ ; c₁ dan c₂ skalar
                    u₂= d₁v₁ + d₂v₂ ; d₁ dan d₂ skalar
$\left [  w\right ]_2 = \begin{bmatrix} k_1 \\ k_2 \end{bmatrix}$

a) Uraikan w = k₁u₁ + k₂ u₂ dan
w = k₁u₁ + k₂ u₂
↔ w = k₁(c₁v₁ + c₂v₂ ) + k₂ ( d₁v₁ + d₂v₂ )
↔ w = ( k₁c₁v₁ + k₁c₂v₂ ) + ( k₂d₁v₁ + k₂d₂v₂ )
↔ w = ( k₁c₁ + k₂d₁) v₁ + ( k₁c₂ + k₂d₂) v₂

Sehingga :
$\leftrightarrow \left [ w \right ]_{2'}=\begin{bmatrix} k_1 c_1&+& k_2d_1\\k_1c_2 &+& k_2d_2
\end{bmatrix}$
$\leftrightarrow \left [ w \right ]_{2'}=\begin{bmatrix} c_1&+& d_1\\c_2 &+& d_2 \end{bmatrix}\begin{bmatrix}k_1\\k_2\end{bmatrix}$
$\leftrightarrow \left [ w \right ]_2=\begin{bmatrix} c_1&+& d_1\\c_2 &+& d_2 \end{bmatrix}\left [ w \right ]_2$


$ \left [ u_1 \right ]_{2'}$                                $\left [ u_2 \right ]_{2'}$

$\therefore \left [ w \right ]_{2'}=\left [ \left [ u_1 \right ]_{2'}\vdots \left [ u_2 \right ]_{2'} \left [ w \right ]_{2}\right ]$
$ P=\left [ \left [ u_1 \right ]_{2'}\vdots \left [ u_2 \right ]_{2'} \right ]$

Disebut matriks transisi dari S ke S’



b) Uraikan w = L₁v₁ + L₂v₂ dan $\left [  w\right ]_{2'} = \begin{bmatrix} L_1 \\ L_2 \end{bmatrix}$
 
w = L₁v₁ + L₂v₂
↔ w = L₁(a₁u₁ + a₂ u₂) + L₂(b₁u₁ + b₂ u₂)
↔ w = (L₁a₁u₁ + L₁a₂ u₂) + (L₂b₁u₁ + L₂b₂ u₂)
↔ w = (L₁a₁ + L₂b₁)u₁ + (L₁a₂ + L₂b₂)u₂

Sehingga :
$\leftrightarrow \left [ w \right ]_{2}=\begin{bmatrix} L_1 a_1&+& L_2b_1\\L_1a_2 &+& L_2b_2
\end{bmatrix}$
$\leftrightarrow \left [ w \right ]_{2}=\begin{bmatrix} a_1&+& b_1\\a_2 &+& b_2 \end{bmatrix}\begin{bmatrix}L_1\\L_2\end{bmatrix}$
$\leftrightarrow \left [ w \right ]_2=\begin{bmatrix} a_1&+& b_1\\a_2 &+& b_2 \end{bmatrix}\left [ w \right ]_{2'}$


$\left [ v_1 \right ]_{2}$                                $\left [ v_2 \right ]_{2}$


$\therefore \left [ w \right ]_{2}=\left [ \left [ v_1 \right ]_{2}\vdots \left [ v_2 \right ]_{2} \left [ w \right ]_{2'}\right ]$
$ P=\left [ \left [ v_1 \right ]_{2}\vdots \left [ v_2 \right ]_{2} \right ]$
            

Disebut matriks transisi dari S’ ke S

Referensi : Catatan Kuliah