Transposes

A good background video

Transposes are inextricably linked to the concept of duality.

The transpose of a vector is its dual vector: essentially represents the operation “project onto ”, and it can be applied to some vector simply by left-multiplying, i.e. . Therefore, should primarily be thought of as a function.

One way to visualize a row vector is by drawing its level curves. That is, for each integer , we draw a single line representing all vectors such that . For instance, here’s how we’d represent the row vector :

The line going through the origin corresponds to , and the line going through and corresponds to .

Note that scaling up the vector will simply make the level curves more dense. Shown below are the level curves for :

Lemma

The distance between the level curves of is .

Now, we’ll use this visualization to prove the following identity geometrically:

Theorem

For any 2x2 matrix , .

As a quick refresher, is the signed area of the parallelogram defined by the column vectors and . For instance, represents the following area:

Equivalently, the determinant is the scale factor by which transforms the area of any shape.

Now, how do we visualize the scaling induced by ? It will actually be easier to visualize this in the backwards direction---that is, to find the area of the shape that maps to the unit square. If this area is , then we hope to show that .

Let’s draw in level curves for . Then, by definition of the level curves, the shape which maps to the unit square is precisely the shaded region:

Now, how can we relate the area of this parallelogram to our first one? Well, according to our lemma, the two altitudes of the new parallelogram are and , respectively. We also know that the two altitudes of the original parallelogram are and , respectively. And since the two parallelograms have the same angles, this means that they must be similar: the new one is just the original one scaled by a factor of ! This means the area of our new parallelogram is simply

as desired.

By the way, the basis which defines the second parallelogram has a special name: it is the dual basis of .

Also unfortunately, this similarity argument doesn’t seem to extend to higher dimensions. However, I hope it still provided a more visceral feel for what exactly the transpose does, and how it’s visually related to the original basis.