How Can I Overcome My Dislike for Linear Algebra Before the Exam?

[mesa]

There, that feels better...

 

[inotyce]

There, that feels better...

 

[dipole]

It's dry at first, but when you see how it starts to tie all the other math together you've learned - differential equations, orthogonal polynomials, Fourier series, and obviously the stuff you do in QM which you may not have realized with linear algebra the whole time... it's actually a pretty cool subject.

Matrices, however, will always be dry to me I think - unless I'm programming them.

 

[lisab]

There, that feels better...

Just curious: did you like Geometry?

 

[UltrafastPED]

IMHO, you can never know too much linear algebra!

 

[mesa]

It's dry at first, but when you see how it starts to tie all the other math together you've learned - differential equations, orthogonal polynomials, Fourier series, and obviously the stuff you do in QM which you may not have realized with linear algebra the whole time... it's actually a pretty cool subject.

Matrices, however, will always be dry to me I think - unless I'm programming them.

From what I have seen there is no doubt about it's usefulness, its the not being able to see what I'm doing that is frustrating :P

I picked up several supplementary texts but they all seem to emphasize proofs of the procedures as opposed to why and how they work. Even Anton's book on the subject was dry, yikes!

Just curious: did you like Geometry?

I love it, favorite subject by far although it seems to be a bit of a lost art today with so much emphasis towards Calculus for geometric derivations (at least for the engineering program at my school). I love what calculus can do but I prefer the old school :)

 

[UltrafastPED]

Linear algebra can be viewed as the theory of vectors (and vector spaces), and their linear transformations ... which are the matrices when you have chosen an explicit basis.

Much of the rest is the mechanics of how to do this, and the conditions that apply.

There are many important applications, from the Schroedinger equation (H |psi> = E |psi>), which is an eigenvalue equation, etc.

 

[ZeroPivot]

what about non-linear algebra =)

 

[mesa]

Linear algebra can be viewed as the theory of vectors (and vector spaces), and their linear transformations ... which are the matrices when you have chosen an explicit basis.

Much of the rest is the mechanics of how to do this, and the conditions that apply.

There are many important applications, from the Schroedinger equation (H |psi> = E |psi>), which is an eigenvalue equation, etc.

I am already convinced of linear algebras usefulness, the problem I am having is we are just learning the basics and no one seems to be able to show how these things work. If on an elementary level it is difficult to 'see' then I fear what is to come.

On the subject of vectors, no doubt trigonometry combined Descartes wonderful coordinates is a powerful combination of techniques by bringing in coordinates to trigonometry. I am routinely amazed at it's usefulness and the insight it provides as this is a whole new way of doing trig for me.

Going just outside the realm of the textbook (Calc III) we can see how vectors can be used for solving certain types of second degree multivariable polynomials, WoW! This is one subject area I will be dedicating much time to during winter break!

It is a shame there isn't more time for exploration during the semester. In the meantime I hold out hope there will be a reasonable tie in with linear algebra at some point...

 

[dkotschessaa]

There seems to be a love hate thing with this subject, and I have a feeling it has a lot to do with the teacher. It's actually a very beautiful subject. The way some of the proofs and identities and things work out - it's very nice and neat. Not messy like calculus. Nice and..well, linear. I don't know how else to say it. don't give up!

 

[D H]

There seems to be a love hate thing with this subject, and I have a feeling it has a lot to do with the teacher.

And the student. It takes a certain kind of person who can appreciate abstract mathematics as a thing of beauty. Other people want, or even need, to see that abstract mathematics be made concrete (see how it is applied) before they can begin understanding it. Those other people probably shouldn't be math majors.

The same goes with the sciences and engineering. I know I've seen posts here by mathematicians who just don't quite grok science. They understand the math with no problem, but the how and why they should use this math or that is a struggle. They found a much better fit to the way they think over in math world.

 

[dkotschessaa]

And the student. It takes a certain kind of person who can appreciate abstract mathematics as a thing of beauty. Other people want, or even need, to see that abstract mathematics be made concrete (see how it is applied) before they can begin understanding it. Those other people probably shouldn't be math majors.

The same goes with the sciences and engineering. I know I've seen posts here by mathematicians who just don't quite grok science. They understand the math with no problem, but the how and why they should use this math or that is a struggle. They found a much better fit to the way they think over in math world.

Yes, true. I ultimately found that i had a love for abstraction, and that applied mathematics was tedious and messy. All those natural forces of the universe kept getting in the way of my pretty equations. so I became a math major.

However, I know a lot of math majors (and at least one teacher who has a masters) who are abstraction-oriented but for some reason still do not like linear algebra. Maybe it's still too 'practical' for them?

-Dave K

 

[nitsuj]

I am already convinced of linear algebras usefulness, the problem I am having is we are just learning the basics and no one seems to be able to show how these things work. If on an elementary level it is difficult to 'see' then I fear what is to come.

I have no idea if this is similar to what you are looking for, but it definitely a "pictograph" of least squares regression to "see" what the math is doing.

http://hspm.sph.sc.edu/courses/J716/demos/LeastSquares/LeastSquaresDemo.html

 

[SW VandeCarr]

I have no idea if this is similar to what you are looking for, but it definitely a "pictograph" of least squares regression to "see" what the math is doing.

http://hspm.sph.sc.edu/courses/J716/demos/LeastSquares/LeastSquaresDemo.html

Hmmm. Is linear regression part of linear algebra? Actually it is discussed in the applications section of a textbook of mine. But it also discusses applications of LA to differential equations.

 

[D H]

Hmmm. Is linear regression part of linear algebra? Actually it is discussed in the applications section of a textbook of mine, But it also discusses applications of LA to differential equations.

It depends. Least squares regression? Absolutely. That least squares is linear is what makes it so easy. On the other hand, a lot of robust estimation techniques are not linear. Many aren't even differentiable (e.g., minimax techniques). Robust estimation is not easy. To make it somewhat tractable, most (all?) robust estimation techniques make locally linear approximations -- and then rinse and repeat.

 

[mesa]

And the student. It takes a certain kind of person who can appreciate abstract mathematics as a thing of beauty. Other people want, or even need, to see that abstract mathematics be made concrete (see how it is applied) before they can begin understanding it. Those other people probably shouldn't be math majors.

I was unaware that abstract math immediately equated to techniques that very few people understand. These processes did not just 'appear' on a sheet of paper, they were made by people who developed them. In this class (and the books I have seen so far) it seems the emphasis is primarily on the technique while the way how these ideas came to fruition are forgotten.

I am a firm believer in that we shouldn't use something unless we understand it. For me this applies to all mathematics, even the basics like being able to derive Pi, e, the quadratic, identities, etc. etc. (and not through memorization but by actual reasoning). If the way of mathematicians today does not follow this path then I am saddened by the state of affairs for such a wonderful subject.

 

[dkotschessaa]

I am a firm believer in that we shouldn't use something unless we understand it.

So much for engineering!

 

[mesa]

So much for engineering!

Hah! :D

 

[D H]

That's a bit naive.

You haven't told us the book, and you haven't even told us the target audience of the class. Is this

• A linear algebra class for math majors,
• A linear algebra course for non-math majors, or
• An applied mathematics class that covers linear algebra along with other stuff (and just when you think you are starting to get one subject the instructor will say "and now for something completely different", switching to integral equations and Green's functions)?

 

[WannabeNewton]

LA is extremely dull until you get to infinite dimensional topological vector spaces. Then stuff gets interesting :) but finite dimensional LA is very boring I agree. The spaces are way too well behaved.

 

[AlephZero]

but finite dimensional LA is very boring I agree. The spaces are way too well behaved.

Maybe they are boring at a conceptual level, but figuring out how to compute stuff efficiently (i.e. fast enough so you can still remember what the question was when you have got the answer, and preferably with more than zero correct significant figures in the answer as well) gets a bit more interesting when the space is say 1,000,000-dimensional rather than 2d or 3d.

 

[lisab]

Just curious: did you like Geometry?

I love it, favorite subject by far although it seems to be a bit of a lost art today with so much emphasis towards Calculus for geometric derivations (at least for the engineering program at my school). I love what calculus can do but I prefer the old school :)

I'm surprised! I absolutely loved geometry too, and linear algebra scratched that same itch.

Linear algebra can be viewed as the theory of vectors (and vector spaces), and their linear transformations ... which are the matrices when you have chosen an explicit basis.

Much of the rest is the mechanics of how to do this, and the conditions that apply.

There are many important applications, from the Schroedinger equation (H |psi> = E |psi>), which is an eigenvalue equation, etc.

Yes, vector spaces and mapping transformations are so visual, that's why I loved it. Being comfortable with LA made quantum feel all warm and cozy .

 

[Pythagorean]

I'm one of those that really needs a physical example in math's and can't think terribly abstractly until after I've seen it in physics. I didn't care much for linear algebra either, but then once I started doing linear stability analysis in dynamical systems, I grew to appreciate it (just a little bit, though, I wouldn't share my ice cream with it).

Didn't appreciate it when I was applying it in QM, because QM, like math, requires abstract thinking. It's really neat using brah-ket notation, though, I guess.

 

[mesa]

That's a bit naive.

Well, excuse me then. This is the general discussion forum and I wasn't really looking for more than a quick vent but since the conversation started rolling I decided to step back in.

Now by your post I am pleasantly surprised to hear I am wrong about linear algebra and I look forward to your interpretation of answers I have been unable to find through my class mates, tutoring center, instructors, etc., etc. I'll send a PM when I post.

 

[jhae2.718]

I do dynamics and controls for aerospace vehicles, so I :!) linear algebra.

From a pure mathematics standpoint it isn't terribly interesting, but it's amazingly useful and should be in everyone's toolbox.

 

[mesa]

...but it's amazingly useful and should be in everyone's toolbox.

From the little I have seen I already agree.

It seems many PFer's have a good 'visual' understanding of linear algebra which means there is a critical piece of information I must be missing. If it is anything like geometry (as lisab suggests) then I'll be in good company.

 

[WannabeNewton]

I'm a geometry lover myself but there's a lot of bias in there because of general relativity :)

 

[mesa]

I'm a geometry lover myself but there's a lot of bias in there because of general relativity :)

Special relativity is wonderful and now I look forward to general relativity. With the new tools from this semester it will be fun to poke around. It must be quite a treat to know it well.

 

[nitsuj]

I am a firm believer in that we shouldn't use something unless we understand it. For me this applies to all mathematics, even the basics like being able to derive Pi, e, the quadratic, identities, etc. etc. (and not through memorization but by actual reasoning). If the way of mathematicians today does not follow this path then I am saddened by the state of affairs for such a wonderful subject.

Please become a math teacher lol

Not so sure about the if you don't understand it don't use it when it comes to equations, but the rest is bang on with my experience.

 

[lendav_rott]

I'm fairly neutral, although I prefer applied math, makes me able to observe the outcome so to speak.
Linear algebra isn't anything horrible, I don't understand what the hate is about - this was also the case during our vector algebra course, some people just DETEST it ..I just..I don't know, I give up..

 

[mesa]

I'm fairly neutral, although I prefer applied math, makes me able to observe the outcome so to speak.
Linear algebra isn't anything horrible, I don't understand what the hate is about - this was also the case during our vector algebra course, some people just DETEST it ..I just..I don't know, I give up..

From the little I have seen I am already impressed by what it can do but using systems where I can't see how they operate is frustrating, memorizing is a terrible way to approach a new subject. I will try my luck with some of our PFer's and hopefully get this resolved.

 

[dlgoff]

After missing a few days of LA lectures at http://www.educator.com/mathematics/linear-algebra/hovasapian/, this thread has has helped me get back on schedule. :thumbs:

 

[mnb96]

mesa,

did you try to have a look at some introduction to Geometric Algebra?
If you are an engineer/computer scientist, you might want to have a look at the first chapters of Dorst's book "Geometric Algebra for Computer Science".

There are chances that your symptoms of "allergy" to Linear Algebra are due to its strong coordinate-dependency. Indeed, working with matrices as blocks of numbers will inevitably obscure very often the underlying geometry of many operations.

In Geometric Algebra you work more abstractly with elements of a vector space which have a very tangible geometric interpretation, and without directly *representing* them with matrices. This is essentially a modern, and coordinate-free approach to the subject.
You will definitely have to master some new algebraic techniques (i.e. Clifford Algebra), but you won't be mechanically manipulating array of numbers or multiplying big matrices.

However, at some point you will learn that matrices (and tensors) are in fact, a convenient representation of elements (and transformations of elements) of a Geometric algebra.

Warning: if you are currently taking a course in linear algebra, and you have time constraints to study, absorb the concepts and pass the exam, then diving into geometric algebra might confuse your ideas, as translating back and forth between the two languages is not a trivial task for a beginner.

 

[mesa]

There are chances that your symptoms of "allergy" to Linear Algebra are due to its strong coordinate-dependency. Indeed, working with matrices as blocks of numbers will inevitably obscure very often the underlying geometry of many operations.

In Geometric Algebra you work more abstractly with elements of a vector space which have a very tangible geometric interpretation, and without directly *representing* them with matrices. This is essentially a modern, and coordinate-free approach to the subject.
You will definitely have to master some new algebraic techniques (i.e. Clifford Algebra), but you won't be mechanically manipulating array of numbers or multiplying big matrices.

However, at some point you will learn that matrices (and tensors) are in fact, a convenient representation of elements (and transformations of elements) of a Geometric algebra.

I will certainly look into this. I checked my local library and they have an (electronic) copy of Dorsts book, I look forward to reading it.

Warning: if you are currently taking a course in linear algebra, and you have time constraints to study, absorb the concepts and pass the exam, then diving into geometric algebra might confuse your ideas, as translating back and forth between the two languages is not a trivial task for a beginner.

Then thank goodness for winter break :)

 

[mnb96]

I will certainly look into this. I checked my local library and they have an (electronic) copy of Dorsts book, I look forward to reading it.

Good. Feel free to come back here and tell us what are your first impressions with Geometric Algebra.
In the meanwhile you can have a look at this paper for an informal and historical introduction to the subject.

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). However keep in mind that all the manipulations with matrices that you saw (and you'll see) are just a coordinate-dependent representation of geometric concepts.

 

[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.