Question 4.7.4: Finding a Transition Matrix Find the transition matrix from ...
Finding a Transition Matrix
Find the transition matrix from B to B´ for the following bases for R³ .
B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} and B´ = {(1, 0, 1), (0, -1, 2), (2, 3, -5)}
First use the vectors in the two bases to form the matrices B and B´ .
B = \left [ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right ] and B´ = \left [ \begin{matrix} 1 & 0 & 2 \\ 0 & -1 & 3 \\ 1 & 2 & -5 \end{matrix} \right ]
Then form the matrix [B´ B] and use Gauss-Jordan elimination to rewrite [B´ B] as [I_3 P^{-1}]
\left [ \begin{matrix} 1 & 0 & 2 & 1 & 0 & 0 \\ 0 & -1 & 3 & 0 & 1 & 0 \\ 1 & 2 & -5 & 0 & 0 & 1 \end{matrix} \right ] ⇒ \left [ \begin{matrix} 1 & 0 & 0 & -1 & 4 & 2 \\ 0 & 1 & 0 & 3 & -7 & -3 \\ 0 & 0 & 1 & 1 & -2 & -1 \end{matrix} \right ]
From this, you can conclude that the transition matrix from B to B´ is
P^{-1} = \left [ \begin{matrix} -1 & 4 & 2 \\ 3 & -7 & -3 \\ 1 & -2 & -1 \end{matrix} \right ] .
Try multiplying P^{-1} by the coordinate matrix of x = [1 2 -1]^T to see that the result is the same as the one obtained in Example 3.