Conversion between 3D Cartesian and skewed coordinate system

Question

The three axes of the skewed coordinate systems are $[1, 0, 0]$, [$1/\sqrt{2}$, $1/\sqrt{2}$, 0] and [$1/\sqrt{3}$, $1/\sqrt{3}$, $1/\sqrt{3}$]. What is the transformation matrix $t_{ij}$ and $g_{ij}$ that transform from Cartesian to the skewed system and from the skewed to the Cartesian respectively. I have read a book entitled mathematical Physics- Applied mathematics for Scientists and Engineering for a few days and still cannot figure it out. Please help! Thank you.

Answer 1

Let consider the matrix

$$M =\begin{bmatrix} 1 & \frac1{\sqrt 2} & \frac1{\sqrt 3}\\ 0 & \frac1{\sqrt 2} &\frac1{\sqrt 3}\\ 0 & 0 & \frac1{\sqrt 2}\end{bmatrix}$$

then $M$ is the transformation matrix from the new coordinate system to the standard cartesian system and $M^{-1}$ is the transformation matrix from the standard cartesian system to the new coordinate system.