\[ \textbf{Change Basis} \]
Given a vector space $U$ with basis $\{u_1, u_2, ... u_n \}$, and other vector space $W$ with basis
$\{w_1, w_2, ... w_n \}$, let $v \in V$, then we have
\begin{equation}
\begin{aligned}
v &=
\begin{bmatrix}
u_1 & u_2 & \cdots & u_n
\end{bmatrix}
\left[ \begin{array}{cc}
c_1 \\
c_2 \\
\vdots \\
c_n
\end{array}
\right] \label{eq:one} \\
\newline
\left[v \right]_U &=
\left[ \begin{array}{cc}
c_1 \\
c_2 \\
\vdots \\
c_n
\end{array}
\right] \mbox{ is the coordinate relative to } U
\end{aligned}
\end{equation}
$\mbox{Since } \{w_1, w_2, ... w_n \} \mbox{ is basis for W}. u_1, u_2, ..., u_n \mbox{ can be represented as} $
\begin{equation}
\begin{aligned}
u_1 &=
\left[w_{1}, w_{2} \,, \cdots \,, w_{n} \right]
\left[ \begin{array}{cc}
a_{11} \\
a_{21} \\
\vdots \\
a_{n1}
\end{array}
\right] \\
u_2 &=
\left[w_{1}, w_{2} \,, \cdots \,, w_{n} \right]
\left[ \begin{array}{cc}
a_{12} \\
a_{22} \\
\vdots \\
a_{n2}
\end{array}
\right] \\
\cdots \\
u_n &=
\left[w_{1}, w_{2} ,\cdots , w_{n} \right]
\left[ \begin{array}{cc}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{nn}
\end{array}
\right] \\
\begin{bmatrix}
u_1 & u_2 & \cdots & u_n
\end{bmatrix} &=
\left[w_1, w_2, \cdots, w_n \right]
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{bmatrix} \nonumber \\
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\mbox{From (1), we have} \\
v &=
\left[w_1, w_2, \cdots, w_n \right]
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{bmatrix}
\left[ \begin{array}{cc}
c_1 \\
c_2 \\
\vdots \\
c_n
\end{array}
\right] \\
\left[ v \right]_{\scriptscriptstyle{W}} &=
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{bmatrix}
\left[ \begin{array}{cc}
c_1 \\
c_2 \\
\vdots \\
c_n
\end{array}
\right] \\
\end{aligned}
\end{equation}
Let $P_{ \scriptscriptstyle{U} \rightarrow \scriptscriptstyle{W}}$ to be the change basis matrix from $V$ to $W$
\begin{equation}
\begin{aligned}
P_{ \scriptscriptstyle{U} \rightarrow \scriptscriptstyle{W} } &=
\begin{bmatrix}
\left[ u_1 \right]_{w} &\left[ u_2 \right]_{w} & \cdots & \left[ u_n \right]_{w}
\end{bmatrix} \\
\left[ v \right]_{\scriptscriptstyle{W}} &=
P_{ \scriptscriptstyle{U} \rightarrow \scriptscriptstyle{W} }
\left[ v \right]_{\scriptscriptstyle{U}} \\
\text{Since } P_{ \scriptscriptstyle{U} \rightarrow \scriptscriptstyle{W} } \mbox{ is invertable matrix} \\
\left[ v \right]_{\scriptscriptstyle{U}} &= P_{ \scriptscriptstyle{U} \rightarrow \scriptscriptstyle{W}}^{-1} \left[ v \right]_{\scriptscriptstyle{W}} \nonumber
\end{aligned}
\end{equation}