Linear transformations are defined by their symmetries: they preserve linear subspaces and linear operations. The associated matrices are used to understand these transformations. Matrices are an algebraic description; via matrices questions of transformations can be turned into algebraic problems. To figure out which points went to zero under a transformation, I showed an algebraic method of calculating the kernel as a solution space. Now that I know the algebraic description of transformations as matrices, I want to investigate what else the matrix can tell me about the transformation. In this section, I am asking two questions in particular.
First, what does the transformation do to the size of objects? I choose the vague word ‘size’ intentionally because I will work in various dimensions. In one dimension, size is the length. In two dimensions, size is area. In three dimensions, size is volume. And in higher dimensions, there is some extended notion of volume that fits that dimension; I can call this hyper-volume. It is important to note that size depends on the ambient dimension. A square in \(\RR^2\) has some area, some non-zero size. But if there is a flat square somewhere in \(\RR^3\text{,}\) it has no volume, therefore no substantial size in that space.
Second, what does the transformation do to the orientation of an object? In
Section 1.2, I defined orientation as a choice of axis directions relative to each other, but that isn’t the most enlightening definition. To help, I will describe orientation for each dimension. In
\(\RR\text{,}\) orientation is a direction: moving in the positive or negative direction along the number line. There are only two directions of movement, so two orientations. If I have a transformation of
\(\RR\text{,}\) I can ask if it changes or preserves these directions. In
\(\RR^2\text{,}\) instead of moving in a line, I think of moving in closed loops or paths. These paths can be clockwise or counter-clockwise. Then I can ask if a transformation changes clockwise loops into other clockwise loops or into counter-clockwise loops. The axis system in
\(\RR^3\) is, conventionally, given by a right-hand-rule. If I know the
\(x\) and
\(y\) directions, the right-hand-rule indicates the positive
\(z\) direction. Then I can ask where these three directions go under a transformation and if a right-hand-rule still applies. If it does, the orientation is preserved. If it doesn’t, and a left-hand-rule would work instead, the transformation reverses orientation. In higher dimensions, there are other extensions of the notion of orientation. In each case, the question is binary: a transformation either preserves or reverses orientation.
