4.4 Determinants, or how linear transformations affect volume
In Section 4.3, we have seen that linear transformations (Definition 16) can be thought of as distorting the grid determined by the basis vectors.
Following our geometric intuition, we suspect that measuring how much a transformation distorts volume and distance can provide some valuable insight. As we will see in this chapter, this is exactly the case. Transformations that preserve distance or norm are special, giving rise to methods such as Principal Component Analysis.
4.4.1 How linear transformations scale the area
Let’s go back to the Euclidean plane one more time. Consider any linear transformation A, mapping the unit square to a parallelogram.
The area of this parallelogram describes how A scales the unit square. Let’s call it λ for now; that is,
where C = [0,1] × [0,1] is the unit square, and A(C) is...
