Breaking Free from Linear Algebra: My Journey of Dislike and Acceptance

[mesa]

Anyways, if you are an undergraduate student, I suggest you to master the classical techniques of linear algebra that you will be taught during the course (they are very useful and you simply can't ignore them).

Although I find the prospect of memorizing mathematics frustrating I have been impressed with the power and versatility of matrices. I hope geometric algebra 'clears' things up.

Good. Feel free to come back here and tell us what are your first impressions with Geometric Algebra.

I certainly will!

 

[Superposed_Cat]

It's my second favorite area of math so far. Calculus being first. I love what math can do, the way everything in the universe can be modeled. It is after all the programming language the universe was written in.

 

[EM_Guy]

For those who hate (or dislike) linear algebra, did you get introduced to determinants early in the course? Were the other concepts of linear algebra then based on the determinant?

I mention this, because I think this is often how linear algebra is presented (in textbooks) and taught, and I think it is absolutely the wrong pedagogical approach.

I managed to get through my undergraduate linear algebra class, passing without learning much. Then, in graduate school, in a course I was taking on fast computational electromagnetics, we were using numerical techniques to find current distributions. My professor took 2 or 3 lectures to review some of the fundamentals of linear algebra (vector spaces, linear independence, span, basis functions, orthogonality, inner products, linear transformations, etc.), and the subject became alive to me I have since bought a textbook called "Linear Algebra Done Right" which develops the subject of linear algebra without using determinants! He includes a chapter on determinants at the end of the book - almost as an unnecessary appendix. He thinks (and I agree) that linear algebra should be taught without determinants.

 

[micromass]

He thinks (and I agree) that linear algebra should be taught without determinants.

And that is exactly why Axler's book is not so good. Like it or not, but determinants are an essential part of linear algebra. It's all very nice and pretty to do prove stuff without them. But once you actually have to calculate something, you'll be very happy that determinants exist. They are an awesomely superior calculation tool. So teaching linear algebra without determinants immediately condemns your students the computational part of linear algebra, which is the main reason they take it to begin with! Aside from computations, determinants have a huge impact in other fields of mathematics, such as projective geometry or differential geometry.

What I do agree with is that they should teach determinants more geometrically (as the "volume" of a figure), or that they shouldn't introduce it immediately in the course. But not introducing it at all is a very big mistake. And this is sadly why I only recommend Axler to students who are already familiar with the computational part of LA. It would not be good as a first book. The book by Treil is infinitely superior to Axler (and it is free!): http://www.math.brown.edu/~treil/papers/LADW/LADW.html

 

[EM_Guy]

And that is exactly why Axler's book is not so good. Like it or not, but determinants are an essential part of linear algebra. It's all very nice and pretty to do prove stuff without them. But once you actually have to calculate something, you'll be very happy that determinants exist. They are an awesomely superior calculation tool. So teaching linear algebra without determinants immediately condemns your students the computational part of linear algebra, which is the main reason they take it to begin with! Aside from computations, determinants have a huge impact in other fields of mathematics, such as projective geometry or differential geometry.

What I do agree with is that they should teach determinants more geometrically (as the "volume" of a figure), or that they shouldn't introduce it immediately in the course. But not introducing it at all is a very big mistake. And this is sadly why I only recommend Axler to students who are already familiar with the computational part of LA. It would not be good as a first book. The book by Treil is infinitely superior to Axler (and it is free!): http://www.math.brown.edu/~treil/papers/LADW/LADW.html

You probably know much more than me, so you probably have a very good point. However, I have written code in MATLAB to solve sets of linear differential equations using the method of weighted residuals - Galerkin method - without any need for a determinant. When it comes to solving problems that require intensive computations, we would either write or use software to solve those equations. And when it comes to real life problems, the number of unknowns becomes so large that the prospect of finding the determinant becomes too big of a task - no?

I'm attaching a paper (with MATLAB) code that I wrote in graduate school in which I solved a non-homogeneous linear differential equation using the MWR - Galerkin method.

So, it seems to me that you can perform computations while still maintaining the concepts undergirding the operations you are performing without determinants by writing code. Whereas, if you use determinants, you are just following an algorithm without understanding what you are doing (or at least that is what it was like for me).

However, I was never taught how to view determinants geometrically. So, maybe that's my problem.

 

[micromass]

You probably know much more than me, so you probably have a very good point. However, I have written code in MATLAB to solve sets of linear differential equations using the method of weighted residuals - Galerkin method - without any need for a determinant. When it comes to solving problems that require intensive computations, we would either write or use software to solve those equations. And when it comes to real life problems, the number of unknowns becomes so large that the prospect of finding the determinant becomes too big of a task - no?

I'm attaching a paper (with MATLAB) code that I wrote in graduate school in which I solved a non-homogeneous linear differential equation using the MWR - Galerkin method.

So, it seems to me that you can perform computations while still maintaining the concepts undergirding the operations you are performing without determinants by writing code. Whereas, if you use determinants, you are just following an algorithm without understanding what you are doing (or at least that is what it was like for me).

However, I was never taught how to view determinants geometrically. So, maybe that's my problem.

The problem with Axler's book is that somebody who finished that and only that would have significant troubles finding eigenvalues to various small matrices. For example, if I were to ask him to compute the eigenvalues to

$$\left(\begin{array}{ccc} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\end{array}\right)$$

then this person would not be able to do this quickly. Do you think that is an acceptable outcome for somebody who took a linear algebra class? I don't think that is acceptable. There are many other examples such as diagonalization, proving that inverting a matrix is a smooth operation, etc.

Of course there are many many methods with matrices that avoid determinants. But I think it is important that somebody who finishes a first LA class knows how to do basic computations. Similarly, somebody who finishes a calculus class, should know how to find basic integrals (while in many real-life computations, the integral rules we use in calculus are not useful either, just like determinants). But we cannot teach all computation methods in a linear algebra class, so determinants become unavoidable.

Whereas, if you use determinants, you are just following an algorithm without understanding what you are doing (or at least that is what it was like for me).

I accept that. But then I'm afraid that your class was just taught badly. That's not really the fault of the determinants though. I'm sure you can also finish a calculus class without knowing what a derivative and an integral is, but being able to compute them with the rules. That doesn't mean we shouldn't teach those rules. It just means the class should emphasize more on intuition and understanding.

Why do we still teach integration by substitution or by parts? Surely we can integrate functions easily nowadays by only using software? The answer is
1) You should be able to find a basic number of examples yourself to get a feel for the process.
2) They are theoretically important rules which are used often in derivations.
The same reasons hold for determinants: they allow you to compute basic examples, and they are useful in theoretical derivations.

 

[phion]

What I do agree with is that they should teach determinants more geometrically (as the "volume" of a figure), or that they shouldn't introduce it immediately in the course.

This is exactly how I was introduced to linear algebra in school after the intermediate rote self-teaching. The volume of a parallelpiped, for example, is simply |u⋅(w×v)|.

 

[EM_Guy]

I'm not sure if my hesitancy to agree with you stems from (a) the way I was taught linear algebra as an undergrad, (b) the way my undergrad text presents the subject - particularly by computing the eigenvalues of a matrix by finding the real roots of the characteristic polynomial of the matrix - a process which was easy enough to do, but which left me with absolutely no understanding of concepts or intuition regarding the nature of eigenvalues and eigenvectors, (c) the fact that in graduate school, my professor demonstrated some extremely powerful and mathematically beautiful numerical techniques to solve systems of non-homogenous differential equations by using the concepts of linear independence, span, and orthogonal basis functions without the use of determinants, (d) the fact that when I went back to study linear algebra independently later, I have always had trouble understanding anything that was derived in terms of determinants, or (e) the fact that Axler has successfully developed linear concepts of linear algebra without using determinants. I suspect that my hesitancy to agree with you stems from all of the above.

After defining eigenvalues and eigenvectors, Axler proceeds in his book to do the following (without determinants or the characteristic polynomial):
1. He proves that if T is a linear operator on a vector space V and if you have m distinct eigenvalues of T corresponding to m nonzero eigenvectors, then the set of eigenvectors are linearly independent.
2. He proves that each operator on V has at most dim V distinct eigenvalues.
3. He proves that every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue.
4. He proves that if a linear operator has an upper triangular matrix with respect to some basis of V, the eigenvalues of the operator consist precisely of the entries on the diagonal of that upper-triangular matrix.
5. He proves that if a linear operator has dim V distinct eigenvalues, then that operator has a diagonal matrix with respect to some basis of V, and equivalently, that the vector space V has a basis consisting of the eigenvectors of the linear operator T.

Contrast this with my undergraduate book / course.
After defining eigenvalues and eigenvectors, before doing anything to explain how these important concepts relate to the concepts of linear independence, span, basis, linear operators, etc., the book immediately defines the characteristic polynomial (in terms of non-intuitive determinants). Then, they prove that the eigenvalues of a matrix are the real roots of the characteristic polynomial of the matrix. This is easy enough to compute, but leaves me with no intuition regarding what I am doing. Then, we are given the procedure (recipe) for diagonalizing a matrix - the first step of which is the find the characteristic polynomial! Nearly everything in the chapter related to eigenvalues and eigenvectors is based on the characteristic polynomial (which depends on the determinant).

I think a course on linear algebra should focus on concepts and developing intuition for the objects of linear algebra - and that these abstract concepts shouldn't be derived based on determinants. Students need to focus instead on span, linear independence, basis functions, linear operators, eigenvalues, eigenvectors, etc. Once they get comfortable with these concepts, then I see no problem with introducing determinants for computational purposes.

 

[micromass]

I think a course on linear algebra should focus on concepts and developing intuition for the objects of linear algebra - and that these abstract concepts shouldn't be derived based on determinants. Students need to focus instead on span, linear independence, basis functions, linear operators, eigenvalues, eigenvectors, etc. Once they get comfortable with these concepts, then I see no problem with introducing determinants for computational purposes.

Then I think we agree. But this is very different from saying that determinants should not be present in the course at all.

 

[EM_Guy]

Then I think we agree. But this is very different from saying that determinants should not be present in the course at all.

But that's what was so frustrating to me in my undergraduate linear algebra experience. Here I am working hard, studying the textbook, trying to make sense of these difficult concepts, but not really getting anywhere, because I was focused so much on determinants. It is certainly true that you can derive nearly everything in linear algebra in terms of determinants, and that was what I was trying to do. No one gave me a heads up that my focus was in the wrong place. Furthermore, the text (and my professor) led me down this road. So, it was a wonderful experience in graduate school and later to start to think about linear algebra without reference to determinants.

I say all this, because I suspect that I'm not the only one whose frustrating experience with linear algebra was for this reason. If people hate linear algebra and struggle "getting it" I think they should really ask themselves, "Is it because I'm trying to understand everything in terms of determinants?"

 

[micromass]

Contrast this with my undergraduate book / course.
After defining eigenvalues and eigenvectors, before doing anything to explain how these important concepts relate to the concepts of linear independence, span, basis, linear operators, etc., the book immediately defines the characteristic polynomial (in terms of non-intuitive determinants). Then, they prove that the eigenvalues of a matrix are the real roots of the characteristic polynomial of the matrix. This is easy enough to compute, but leaves me with no intuition regarding what I am doing. Then, we are given the procedure (recipe) for diagonalizing a matrix - the first step of which is the find the characteristic polynomial! Nearly everything in the chapter related to eigenvalues and eigenvectors is based on the characteristic polynomial (which depends on the determinant).

How I would do it is first to introduce the determinant as a measure for invertibility of a matrix. This can easily be seen from the "definition" of a determinant as the volume of a parallelepiped. If the matrix is not invertible, then this parallelepiped will collapse to a subspace and will have volume 0. So the determinant being 0 means that the matrix is not invertible.
Now consider an eigenvalue $\lambda$ of a linear map $T$. Then we have $Tx=\lambda x$ for some nonzero $x\in V$. And hence $(T-\lambda I)x = 0$. We immediately get that the map $T-\lambda I$ is not invertible, and by a miraculous theorem (rank nullity) we get that the non-invertibility of $T-\lambda I$ is the same as the existence of a vector $x$ such that $(T-\lambda I)x = 0$. Now to test non-invertibility, we have seen that we can use the determinant. So $\lambda$ is an eigenvalue iff $T-\lambda I$ is not invertible iff $\text{det}(T-\lambda I) = 0$. So we introduce the characteristic polynomial $\text{det}(T-xI)$, and we have seen that the zeroes of that correspond exactly to the eigenvalues.
Then I would show different ways of viewing the determinant such as the product of all eigenvalues (which makes it correspond nicely to the trace which is the sum of all eigenvalues).

I don't think this should be too difficult for the students?

 

[nuuskur]

Oh god, I have an exam next week on linear algebra.