As we saw in the previous chapter, linear transformations can be represented as matrices. Given that square matrices can be singular or nonsingular, linear transformations under square matrices also have singular and nonsingular properties.
Matrices are widely used for specifying geometric transformation because linear transformation allows for an efficient way of warping points around in a vector space. In fact, the determinant of a square matrix determines how the orientation and volume in the n-dimensional space is scaled. For example, in a 2-d space,
We can see that the nonsingular (nondegenerate) square matrix
$$ A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix} $$
performs a linear transformation that scales the area of the unit square by a multiple of its determinant $det(A) = 5$
However, when the linear transformation is represented by a singular matrix (which means the matrix has linearly dependent rows), **** we can see that the resulting transformation has zero area in the 2-dimensional space. Therefore, singular transformations map the original vector space to a lower dimension and the determinant of the matrix is equal to 0.
In the 3-dimensional space, a linear transformation represented by a nonsingular matrix transforms the geometry of the unit cube to a parallelepiped and the volume of this parallelepiped gives us the determinant of the matrix.
The n-dimensional nonsingular matrix transforms the unit n-cube to a n-dimensional parallelotope.