What is a determinant?

academics linear algebra mathematics
By Elia

This is a blog post about understanding linear maps and a special number associated to them called the determinant. A linear map f from Rn (n-dimensional Euclidian space) to Rm (m-dimensional Euclidian space) is any map which satisfies the following properties:

Screen Shot 2023-11-07 at 4.28.57 PM

We are sometimes taught to think of a linear map as a function defined using coordinates. For example, here is one linear map:

Screen Shot 2023-11-07 at 4.29.22 PMOr we are taught to think of a linear map as a matrix. Here is the matrix that corresponds to the linear map above:

Screen Shot 2023-11-07 at 4.29.42 PM

A corresponds to f because for any vector v in R2 we have:

Screen Shot 2023-11-08 at 1.08.24 PM

Another important way of understanding linear maps, which I want to introduce in this blog post, is through pictures. Here is a picture that helps us understand the map f above.

Screen Shot 2023-11-07 at 4.30.17 PMFigure 1. Grid drawn at scale 1.

If we call the set of points inside the square on the left, S, and we call the set of points inside the parallelogram on the right, R, then the image of S under the map f is R. In other words:

Screen Shot 2023-11-07 at 4.30.40 PM

The picture also shows us how the axes of S, which are drawn in different colors, are mapped under S to the axes of R. For instance, the purple vector on the left is 0,1 and it gets mapped to the purple vector on the right, which is 1,1.

In higher dimensions, we can no longer draw pictures and we’ll need to rely on the coordinate or matrix representations of linear maps, but we can still get a lot of intuition from looking at these kinds of pictures in 2 and 3 dimensions.

Now let’s introduce the determinant.

The determinant is a number associated to a linear whose range and domain have the same dimension.

Suppose f is any linear map from Rn to Rn. The determinant of f is a real number, which we denote by det (f) , and is defined by the following equation:

Screen Shot 2023-11-08 at 2.10.05 PM

where Ω is any region in Rn of positive volume, and vol(Ω) stands for the n-dimensional volume of Ω.

It should be surprising that there is one number that makes this equation true no matter which region we choose!

For the interested reader, we’ll explain why this is so in the appendix, but for now let’s just take this fact for granted.

An example:

Let’s try this definition out on our previous example. Choose Ω to be our square S from Figure 1. The domain and range are 2-dimensional, so n=2 and vol(S) is just the area of S which is 4.

Screen Shot 2023-11-08 at 1.31.09 PM

We can find the area of f(S), which we saw is R, by adding up the areas of the different colored triangles in Figure 2.

vol(purple triangle) = vol(green triangle) = 6

vol(red triangle) = vol(orange triangle) = 2

Adding all these up, we get vol(R) = 16, and so: vol(f(S)) = 4vol(S)

Finally, from our definition of the determinant we see that f = 4.

A surprising property of determinants...

In this section, we’ll discuss a surprising property of determinants. Suppose we have another linear map g. We let g º f stand for the composition of g and f. In other words,

Screen Shot 2023-11-08 at 2.14.48 PM

What can we say about det(g º f)? From our definition of the determinant we have,

Screen Shot 2023-11-08 at 2.14.57 PM

But since we are free to choose f(Ω) as the region in our definition for the determinant of g, we get that:

Screen Shot 2023-11-08 at 2.15.06 PM

Putting the two together we find,

Screen Shot 2023-11-08 at 2.15.13 PM

If g corresponds to a matrix B and f corresponds to a matrix A, this equation becomes

Screen Shot 2023-11-08 at 2.15.21 PM

So, when we compose matrices, the determinants just multiply!

A formula for the determinant of a 2x2 matrix

A formula we often learn first in a lecture about determinants is

Screen Shot 2023-11-08 at 2.19.12 PM

Let’s see how we might derive this from the definition we have.

The matrix above corresponds to the map:

Screen Shot 2023-11-08 at 2.19.22 PMHere is a picture for how the map f transforms the region Ω = square at the origin with side-length 1.

Screen Shot 2023-11-08 at 2.19.34 PM

To find det⁡(f) we need to compute vol(f(Ω)). To do this take a look at the following picture.

Screen Shot 2023-11-08 at 2.19.55 PM

vol(f(Ω)) is just the area of the rectangle with sides a+b and c+d, minus the area of all the colored regions in Figure 4. So, we just need to compute the areas of the colored regions.

Screen Shot 2023-11-08 at 2.20.06 PMPutting this together we have,

Screen Shot 2023-11-08 at 2.20.15 PM

And we get our formula:

Screen Shot 2023-11-08 at 2.20.21 PM

Appendix

In this appendix, we discuss the surprising fact mentioned after the definition of the determinant. It’s a more advanced section and we’ll assume some familiarity with the concept of a proof as well as the concept of a limit. In particular, we’ll prove the following.

Theorem: Suppose f is any linear map from R2 to R2. Then there is a unique real number called det⁡(f) for which the following equation is true,

Screen Shot 2023-11-08 at 2.27.20 PM

whenever Ω is a region in R2 of positive volume.

(The statement is true if R2 is replaced with Rn and the proof is almost exactly the same.)

Screen Shot 2023-11-08 at 2.28.40 PM

Notice that we have:

Screen Shot 2023-11-08 at 2.29.11 PM

Screen Shot 2023-11-08 at 2.29.20 PM

For example, in Figure 5, the region shaded in orange is and the blue line outlines all the copies of the square (1/i)S that cover it. The copies of S can be visualized as the individual cells from the grid in Figure 5.

Now notice that as i goes to infinity, the approximation of our region by the copies of S(i) gets better and better. In particular, as i goes to infinity we see that vol(S(i))N(i) approaches vol(Ω).

From the additivity and scaling property of linear maps we can see that:

Screen Shot 2023-11-08 at 2.31.40 PM

And from this it follows that:

Screen Shot 2023-11-08 at 2.31.47 PM

Notice that we have shown that this volume doesn’t depend on k. Now see let’s what this says about vol(f(Ω)). As i goes to infinity, our approximation of Ω by copies of (1/i)S gets better and better. So, we get the following limit:

Screen Shot 2023-11-08 at 2.31.54 PM

Since all the copies are disjoint we have

Screen Shot 2023-11-08 at 2.32.02 PM

Combing this we the above limit we get,

Screen Shot 2023-11-08 at 2.35.38 PM

Notice also that, as i goes to infinity, our approximation of f(Ω) by the images of our copies under f, gets better and better. From this we get the following limit

Screen Shot 2023-11-08 at 2.35.49 PM

Since all the copies are disjoint we have

Screen Shot 2023-11-08 at 2.35.59 PM

Putting everything together we get

Screen Shot 2023-11-08 at 2.36.06 PM

Which finishes our proof.

Elia graduated from the University of Chicago with a BS in Mathematics and was awarded the Paul R. Cohen prize for achieving the one of the highest academic records in the field of mathematics in his class. Currently, he is completing a PhD in Mathematics at MIT.

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