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Example $1 :$ Let $V$ be $R^{3}$ and let $S=\left\{v_{1}, v_{2}, v_{3}\right\}$ and $. T=\left\{w_{1}, w_{2}, w_{3}\right\}$ be bases for $R^{3},$ where:

$\begin{array}{lll}{v_{1}=(2,0,1)} & {v_{2}=(1,2,0)} & {v_{3}=(1,1,1)} \\ {w_{1}=(6,3,3)} & {w_{2}=(4,-1,3)} & {w_{3}=(5,5,2)}\end{array}$

a) compute the transistion matrix $P_{s\leftarrow t}$ from the $T-$ basis to the $S-$ Basis
a) Verify equation 1 for $v=(4,-9,5)$

$w_{1}=a_{1} v_{1}+a_{2} v_{2}+a_{3} v_{3}$
$w_{2}=b_{1} v_{1}+b_{2} v_{2}+b_{3} v_{3}$
$w_{3}=c_{1} v_{1}+c_{2} v_{2}+c_{3} v_{3}$

$\left(\begin{array}{lll|l}{2} & {1} & {1} & {6} \\ {0} & {2} & {1} & {3} \\ {1} & {0} & {1} & {3}\end{array}\right)$

$\Rightarrow a_{1}, a_{2}, a_{3}$

$\left(v_{1} v_{2} v_{3} | w_{1} : w_{2} : w_{3}\right)$

$\left(\begin{array}{lll|lll}{2} & {1} & {1} & {6} & {4} & {5} \\ {0} & {2} & {1} & {3} & {-1} & {5} \\ {1} & {0} & {1} & {3} & {3} & {2}\end{array}\right)$

$\Rightarrow\left(\begin{array}{lll|ll}{1} & {0} & {0} & {2} & {2} & {1} \\ {0} & {1} & {0} & {1} & {-1} & {2} \\ {0} & {0} & {1} & {1} & {1} & {1}\end{array}\right)$

$P_{S \leftarrow {T}}=\left(\begin{array}{ccc}{2} & {2} & {1} \\ {1} & {-1} & {2} \\ {1} & {1} & {1}\end{array}\right)$

eq 1

$[V]_{S}=P_{S \leftarrow T}\left[V]_{T}\right.$

$[V]_{T} \Rightarrow V=a_{1} w_{1}+a_{2} w_{2}+a_{3} w_{3}$

$\Rightarrow[V]_{T}=\left[\begin{array}{c}{1} \\ {2} \\ {-2}\end{array}\right]$

$[V]_{S} \Rightarrow V=b_{1} w_{1}+b_{2} w_{2}+b_{3} w_{3}$

$[V]_{s}=\left[\begin{array}{c}{4} \\ {-5} \\ {1}\end{array}\right]$

$P_{s \leftarrow T}=\left(\begin{array}{ccc}{2} & {2} & {1} \\ {1} & {-1} & {2} \\ {1} & {1} & {1}\end{array}\right)$

$P_{s \leftarrow T}[V]_{T}=\left(\begin{array}{ccc}{2} & {2} & {1} \\ {1} & {-1} & {2} \\ {1} & {1} & {1}\end{array}\right)\left(\begin{array}{c}{1} \\ {2} \\ {-2}\end{array}\right)=\left(\begin{array}{c}{4} \\ {-5} \\ {1}\end{array}\right)=[V]_{s}$