Cartesian coordinates and Linear Transformation

Question

Equation 3.1(a) is a linear transformation but what is the meaning of 3.1(b) and 3.1(c)? Why should it satisfy these conditions?

Answer 1

Suppose for the moment that $b_i=0$ for $1\le i\le n$ and take for convenience $n=2.$ Then, eq. $3.1a$ just says that

$\overline x_1=a_{11}x_1+a_{12}x_2$ and $\overline x_2=a_{21}x_1+a_{22}x_2.$ In matrix form, this is

$\begin{pmatrix} a_{11} &a_{12} \\ a_{21}& a_{22} \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}=\begin{pmatrix} \overline x_1\\ \overline x_2 \end{pmatrix}\tag1 $

Eq $3.1b$ says that the rows are orthonormal and eq $3.1c$ says that the determinant of the foregoing $2\times 2$ matrix is zero, which you can show by direct calculation.

In particular, this means that $a_{11}^2+a_{12}^2=1$ and $a_{21}^2+a_{22}^2=1$ so remembering the identity $\sin^2{\theta}+\cos^2{\theta}=1$, we can solve the preceding two equations for $\theta$ by setting $x=\cos\theta$ and $y=\sin\theta$. The intuition for this follows from an analysis of the diagram

The case $b_i\neq 0$ for some $1\le i\le 2$ follows at once from the above analysis, by translating the origin from $(0,0)$ to $(b_1,b_2).$

Now it should be clear how to extend this to the three dimensional case.