Solution:

Co-ordinate frames:

Let $\mathrm{p}$ be a vector in $\mathrm{Rn}$

Let $\mathrm{X}=\{\mathrm{x} 1, \mathrm{x} 2, \mathrm{x} 3, \ldots \mathrm{xn}\}$ be a complete orthonormal set for $\mathrm{Rn}$.

The coordinates of p w.r.t X are denoted as $[p]^X$

Can be defined as:

$ p=\sum_{k=1}^n[p]_k^X x^k \\ $

Coordinate Transformation:

$ \begin{aligned}\\ &F=\{f 1, f 2, f 3, \ldots f n\} \\\\ &M=\{m 1, m 2, m 3, \ldots . m n\}\\ \end{aligned}\\ $

F and M are coordinate frames for $\mathbf{R n}$

$ \begin{aligned}\\ {[P]_k^F } &=P . f^k\left(k^{\text {th }} \text { coordinate of } P \text { w.r.t. frame } F\right) \\\\ &=\left(\sum_{j=1}^n[P]_j^M m^j\right) \cdot f^k \\\\ &=\sum_{j=1}^n[P]_j^M\left(m^j \cdot f^k\right) \\\\ &=\sum_{j=1}^n\left(m^j \cdot f^k\right)[P]_j^m \\\\ &=\sum_{j=1}^n\left(f^k \cdot m^j\right)[P]_j^m \\\\ &=\sum_{j=1}^n A_{k j}[P]_j^m \\\\ \therefore[P]^F &=A[P]^{m 1}\\ \end{aligned}\\ $

Inverse Coordinate Transformation:

$\mathrm{F}$ and $\mathrm{M}=$ orthonormal coordinate frames in $\mathrm{Rn} $

$\mathrm{A}=\mathrm{CTM}$ maps $\mathrm{M}$ coordinates into $\mathrm{F}$ coordinates

$\mathrm{A}^{-1}=\mathrm{A}^{\mathrm{T}} $

$ \begin{aligned}\\ \left(A^{-1}\right)_{k j} &=m^k \cdot f^j \\\\ &=f^j \cdot m^k \\\\ &=A_{j k} \\\\ &=\left(A^T\right)_{k j}\\ \end{aligned}\\ $