PX153 - K14 - basis changes and similarity transformations

\begin{aligned} {\vec{e}}^{'} & = S_{11} {\vec{e}}_{1} + S_{12} {\vec{e}}_{2} + S_{13} {\vec{e}}_{3} \\ [\begin{array}{c} {\vec{e}}_{1}^{'} \\ {\vec{e}}_{2}^{'} \\ {\vec{e}}_{3}^{'} \end{array}] & = [\begin{array}{c} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{array}] [\begin{array}{c} {\vec{e}}_{1} \\ {\vec{e}}_{2} \\ {\vec{e}}_{3} \end{array}] \end{aligned}

\begin{aligned} {\vec{r}}^{'} & = x_{1}^{'} {\vec{e}}_{1}^{'} + x_{2}^{'} {\vec{e}}_{2}^{'} + x_{3}^{'} {\vec{e}}_{3}^{'} \\ = \sum_{i = 1}^{3} x_{i}^{'} {\vec{e}}_{i}^{'} \\ = \sum_{i = 1}^{3} x_{i}^{'} \sum_{j = 1}^{3} S_{i j} {\vec{e}}_{j} \\ = \sum_{j = 1}^{3} (\sum_{i = 1}^{3} x_{i}^{'} S_{i j}) {\vec{e}}_{j} \\ l e t, x_{j} & = \sum_{i} x_{i}^{'} S_{i j} \\ {\vec{r}}^{'} = \sum_{j} x_{j} {\vec{e}}_{j} & = \sum_{i} x_{i}^{'} {\vec{e}}_{i}^{'} \end{aligned}

\begin{aligned} [\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}] & = S [\begin{array}{c} x_{1}^{'} \\ x_{2}^{'} \\ x_{3}^{'} \end{array}] \\ [\begin{array}{c} x_{1}^{'} \\ x_{2}^{'} \\ x_{3}^{'} \end{array}] & = S^{- 1} [\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}] \\ b u t, [\begin{array}{c} {\vec{e}}_{1}^{'} \\ {\vec{e}}_{2}^{'} \\ {\vec{e}}_{3}^{'} \end{array}] & = S^{- 1} [\begin{array}{c} {\vec{e}}_{1} \\ {\vec{e}}_{2} \\ {\vec{e}}_{3} \end{array}] \end{aligned}

\begin{aligned} [\begin{array}{c} {\vec{e}}_{1}^{'} \\ {\vec{e}}_{2}^{'} \end{array}] & = [\begin{array}{c} 2 & 0 \\ 0 & 2 \end{array}] [\begin{array}{c} {\vec{e}}_{1} \\ {\vec{e}}_{2} \end{array}] \\ [\begin{array}{c} x_{1}^{'} \\ x_{2}^{'} \end{array}] & = [\begin{array}{c} 1 / 2 & 0 \\ 0 & 1 / 2 \end{array}] [\begin{array}{c} x_{1} \\ x_{2} \end{array}] = [\begin{array}{c} \frac{x_{1}}{2} \\ \frac{x_{2}}{2} \end{array}] \end{aligned}

- basis is two times larger, so the vectors will be two times smaller in the new basis