This transformation squishes all of space onto a line or onto a single point.
Determinant can get negative number.
Negative determinant invert the orientation space
Originally j hat is left side of i hat. but inverted by transformation that has nagative determinant, then j hat is right side of i hat. → Orientation space is inverted.
If we draw graph that shows determinant changes from positive to negative, space will be squished, then will be line, then will be flipped.
Determinant formular is ad-bc. we can get this formular using vector area diagram.
In three dimension, this idea is maintained.
Determinant informs the volume of basis vectors and if it is zero, then it means column vectors must be linear dependent.
We can define basis direction by right hand rule.
index finger : i hat
midde finger : j hat
thumb : k hat
If Determinant is smaller than zero, we have to change right hand rule to using left hand(Orientation flipped)
\[det\left ( \begin{bmatrix}a & b &c\\ d & e &f \\ g&h&i\end{bmatrix} \right ) = a \cdot det\left ( \begin{bmatrix}e & f\\ h & i\end{bmatrix} \right ) - b\cdot det\left ( \begin{bmatrix}d & f\\ g & i\end{bmatrix} \right ) + c\cdot det\left ( \begin{bmatrix}d & e\\ g & h\end{bmatrix} \right )\]