$\textbf{Change Basis}$ 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{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} $\mbox{Since } \{w_1, w_2, ... w_n \} \mbox{ is basis for W}. u_1, u_2, ..., u_n \mbox{ can be represented as}$ \begin{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} \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} Let $P_{ \scriptscriptstyle{U} \rightarrow \scriptscriptstyle{W}}$ to be the change basis matrix from $V$ to $W$ \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}