[Linear Algebra] 5. The determinants

Most of linear transformations stretch or squish space. So measuring how much the space is stretched or squished could be very useful for understanding linear transformation.

Determinant

Because of linear transformation properties, it's possible to predict "how areas in space change", if you know "how much the area of single unit square changes".

Factor representing "how much the area of single unit square changes" is called the determinant of the transformation.

 

For example, 1). the determinant of a transformation would be $3$ if transformation increases the area of the region by a factor of $3$. 2). The determinant of a transformation would be $1/2$ if it squishes areas by a factor of $1/2$. And, 3). the determinant of a 2-D transformation is $0$ if it squishes plane onto a line or even onto a single point. Last example is pretty important. It means that if the determinant of a given matrix is $0$, then transformation associated with that matrix squishes everything into a smaller dimension. Moreover, if orientation of space is inverted, the determinant will be negative. Any transformation that flips space is said to "invert the orientation of space".

 

Let's expand to 3-D. It tells how much volume gets scaled. A determinate of $0$ would mean that all of space is squished onto something with $0$ volume.(e.g., flat plane, line, or single point)

 

Determinant $0$ means that the columns of the matrix are linearly dependent.

 

The formula of determinant:

$$
\det\left(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\right)
=
ad - bc
$$