College-Level Math Options for High School Students
This insight is written for high school students who don’t feel very challenged by their high school courses or who want a taste of what university mathematics is like. You don’t have to wait until university to see exciting mathematics—you can learn it now. Of course, some topics require heavier prerequisites (for example, multiple calculus courses), so you won’t be able to do everything immediately. However, there are still many accessible options.
Do not neglect your high school courses. I get it: factorizing 50 polynomials for homework is pretty boring. Those skills are important and you need to master them or you will get in trouble. Do not think that being able to do university mathematics somehow lets you skip high school math. It doesn’t.
You should also seriously consider mathematics competitions. They can be great fun if you enjoy that sort of thing. I personally never was, and I failed every single competition badly. So yes, it is possible to be successful in mathematics without liking or excelling at competitions. Competitions are not the same as university mathematics; they’re a separate activity.
So what are some things you could do? Here are a few examples (there are many more).
Table of Contents
1) Abstract algebra
This subject typically does not require calculus or trigonometry. Some examples or exercises may use those notions, but not in any deep way (you can usually skip them without harm). A solid introductory book is Pinter’s “A Book of Abstract Algebra“. See also http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178. Armstrong’s “Groups and Symmetry” offers a more geometric approach: http://www.amazon.com/Groups-Symmetry-Undergraduate-Texts-Mathematics/dp/0387966757.
So what should you know?
- You should be acquainted with solving polynomial equations; the quadratic formula is a must.
- You should know basic set theory: the language of sets, functions, unions, intersections, and Cartesian products.
- You should be comfortable with elementary number theory: prime numbers and the fundamental theorem of arithmetic.
- You should know basic proof methods, for example proof methods like mathematical induction and contradiction.
- Some familiarity with ruler-and-compass constructions is helpful but not required.
- In geometry, be comfortable with vectors and equations of lines and planes.
What will you learn?
- Group theory, which describes symmetries: from symmetries of geometric objects (like cubes) to symmetries in physical theories.
- Ring theory, which has applications in coding theory and algebraic geometry. In ring theory you will study the fundamental theorem of arithmetic more deeply.
- Field theory, which leads to a deep study of polynomial equations and explains why most polynomial equations do not have solutions expressible by simple formulas.
2) Linear algebra
Linear algebra is geometry in arbitrary dimensions: you will see what points, lines, and planes look like in very high-dimensional spaces. Many methods of linear algebra are central to science and engineering, so this is an applicable subject. I recommend MacDonald’s “Linear and Geometric Algebra” for an accessible introduction to geometric algebra: http://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932.
What should you know?
- Be very comfortable with vectors, vector addition, and the inner (dot) product as seen in geometry.
- Be familiar with matrices: solving linear equations using Gaussian elimination, multiplying matrices, and inverting matrices.
- Some familiarity with proof techniques and basic set theory is helpful.
What will you learn?
- The abstract notion of a vector space, which appears throughout physics and engineering.
- Geometric algebra, a useful but often overlooked framework for geometric computations.
- The theory of linear maps and matrix diagonalization, which are widely used tools.
3) Euclidean geometry
Why not learn from the masters? Euclid’s Elements remains one of the most important math books ever written. I highly recommend the online version: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html. If you prefer a modern edition, try Casey and Callahan’s “Euclid’s Elements Redux”: http://www.amazon.com/Euclids-Elements-Redux-Daniel-Callahan/dp/1312110783. For more advanced commentary, see the first chapter of Hartshorne’s “Geometry: Euclid and Beyond” (later chapters may require abstract algebra): http://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/0387986502.
What do you need to know?
- You don’t need specific prerequisites—just motivation and willingness to work through the material.
What will you learn?
I recommend working through at least the first four books of Euclid. These develop geometry from first principles and culminate in classical constructions such as the regular pentagon—an elegant and instructive result.
4) Affine, projective, spherical, and hyperbolic geometry
I encountered these topics in high school and they were decisive for my interest in mathematics. I recommend Brannan, Esplen, and Gray’s “Geometry”. These topics require somewhat more prerequisites than typical high school courses, but the gap is bridgeable.
What do you need to know?
- A good knowledge of Euclidean geometry, including vector computations and equations of lines and planes.
- Solid skills with matrix computations: multiplying matrices, inverting matrices, diagonalization, and solving linear systems. If you haven’t seen diagonalization, introductory videos (for example, Khan Academy) can help you learn the basics.
- Algebra skills for solving polynomial equations.
- Some very basic group theory. The first few chapters of Pinter or Armstrong are sufficient; familiarity with groups and subgroups is useful.
What will you learn?
- Conic sections and their applications, including practical properties of parabolas.
- Projective geometry, which is both beautiful and useful (for example, in image processing).
- Spherical geometry and its many applications.
- Hyperbolic geometry, a strikingly different geometry that has important connections to relativity.
5) Other
There are many other subjects and books worth exploring. An honorable mention is John Stillwell’s work, such as “The Four Pillars of Geometry” and “Roads to Infinity”. Stillwell is an excellent expositor, and there is likely a topic of his that will appeal to you.
Advanced education and experience with mathematics
”
Techniques such as combinatorial Nullstellensatz are great help when solving certain olympiad problems.
Here is a problem from a past TST (one of the high school proofwriting math competitions).
For a prime [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG], a subset [IMG]http://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png[/IMG] of residues modulo [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG] is called a sum-free multiplicative subgroup of [IMG]http://latex.artofproblemsolving.com/b/1/e/b1e3fe5347b6585e969d11f1b9c9c6e4e9b52a44.png[/IMG] if
[IMG]http://latex.artofproblemsolving.com/d/5/3/d534b942086dae501cbdce030206adb87567bb07.png[/IMG] there is a nonzero residue [IMG]http://latex.artofproblemsolving.com/1/0/f/10f32377ac67d94f764f12a15ea987e88c85d3e1.png[/IMG] modulo [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG] such that [IMG]http://latex.artofproblemsolving.com/d/a/c/dac8a278d33f59792b9ca46a992a6479675cec01.png[/IMG] (all considered mod [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG]), and
[IMG]http://latex.artofproblemsolving.com/d/5/3/d534b942086dae501cbdce030206adb87567bb07.png[/IMG] there are no [IMG]http://latex.artofproblemsolving.com/4/8/1/481f18278a02eb4c8a5e0ef690f77801ceda1bd8.png[/IMG] (not necessarily distinct) such that [IMG]http://latex.artofproblemsolving.com/8/b/4/8b4b1f26d239c4248e00ab2f36aead778663960e.png[/IMG].
Prove that for every integer [IMG]http://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png[/IMG], there is a prime [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG] and a sum-free multiplicative subgroup [IMG]http://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png[/IMG] of [IMG]http://latex.artofproblemsolving.com/b/1/e/b1e3fe5347b6585e969d11f1b9c9c6e4e9b52a44.png[/IMG] such that [IMG]http://latex.artofproblemsolving.com/c/5/c/c5cde33941eef066d2893edbd828b0e9d7d1d55a.png[/IMG].
Sorry for the large text (I don’t know how that happened, something wrong with the copy and paste).”
I never heard of the combinatorial nullstellensatz before neither did I ever need it. This is why I wrote my insight, I wanted to present people books and literature that gave you the knowledge that actual mathematicians study and that people will definitely encounter and need in university. Not special kind of techniques that are pretty obscure outside the context of competitions.
”
Where are you going to school that allows you to proceed so quickly to college-level math? :wideeyed:
”
My area in general does not have many students progressing forward quickly in math (actually the only reason I even know many extremely advanced students is through AoPS). My situation was honestly just pure luck in that my school happened to recognize that I was advanced enough to complete Algebra 2, Precalc, AP Calc BC, and Calculus III all in the same year (spent hours everyday in the summer studying). I suppose my school had never seen such a student before, and it got excited.
“The majority of high school students that I know that are “interested in math” (not quite sure what you mean by that – do you mean people who enjoy math or people who are seriously thinking about math/physics as a career?) do not know Projective Geometry, Graph Theory, and Abstract Algebra.”
Well, in my actual city (there is a city nearby which has much better standards) math (and actually also physics) is looked down upon by nearly everyone (at least as far as I can see), and the only other kid I know who is seriously thinking about persuing math in the future is definitely better (in that he knows more in depth) than me in nearly all areas of math, however, his school won’t let him skip AP Calc BC.
The statement about proofs was a reference to high school math proofwriting competitions such as the TST and IMO which often require in-depth knowledge of topics in Group Theory and Abstract Algebra.
“I am a bit in awe since I have no opportunity to attend a Diff EQ class right now (as a sophomore) and being in Geometry in 9th grade is the highest you can normally do in my school (and other schools around my area).”
I think most areas have similar rules.
There was a similar rule in my middle school, however my high school has something called a “test-out” in which you can essentially take the final exam of a class before you take the class, and if you score high enough (80% is the bare minimum), then you are allowed to skip the class. There was a similar idea mentioned at a college I tried this in, but they wanted the fees for the class in exchange for the exam, and they would not give me credit.
“But it’s not fair to say there are many high school students with a good grasp on projective geometry and abstract algebra. I really don’t know where you get this from.”
In retrospect I can agree that the use of the term “many” was quite vague. There are just a few. (Try being on AoPS all day every day when almost everybody seemingly knows everything :-p)
“Also, abstract algebra is a huge field. It takes years before you really get the basics of it. Do you really mean to say that high school students know the Sylow theorems? The fundamental theorem of Galois correspondence? The Nullstellensatz? The chinese remainder theorem? etc. Somehow I find this very very very very hard to believe…”
Techniques such as combinatorial Nullstellensatz are great help when solving certain olympiad problems.
Here is a problem from a past TST (one of the high school proofwriting math competitions).
For a prime [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG], a subset [IMG]http://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png[/IMG] of residues modulo [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG] is called a sum-free multiplicative subgroup of [IMG]http://latex.artofproblemsolving.com/b/1/e/b1e3fe5347b6585e969d11f1b9c9c6e4e9b52a44.png[/IMG] if
[IMG]http://latex.artofproblemsolving.com/d/5/3/d534b942086dae501cbdce030206adb87567bb07.png[/IMG] there is a nonzero residue [IMG]http://latex.artofproblemsolving.com/1/0/f/10f32377ac67d94f764f12a15ea987e88c85d3e1.png[/IMG] modulo [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG] such that [IMG]http://latex.artofproblemsolving.com/d/a/c/dac8a278d33f59792b9ca46a992a6479675cec01.png[/IMG] (all considered mod [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG]), and
[IMG]http://latex.artofproblemsolving.com/d/5/3/d534b942086dae501cbdce030206adb87567bb07.png[/IMG] there are no [IMG]http://latex.artofproblemsolving.com/4/8/1/481f18278a02eb4c8a5e0ef690f77801ceda1bd8.png[/IMG] (not necessarily distinct) such that [IMG]http://latex.artofproblemsolving.com/8/b/4/8b4b1f26d239c4248e00ab2f36aead778663960e.png[/IMG].
Prove that for every integer [IMG]http://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png[/IMG], there is a prime [IMG]http://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png[/IMG] and a sum-free multiplicative subgroup [IMG]http://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png[/IMG] of [IMG]http://latex.artofproblemsolving.com/b/1/e/b1e3fe5347b6585e969d11f1b9c9c6e4e9b52a44.png[/IMG] such that [IMG]http://latex.artofproblemsolving.com/c/5/c/c5cde33941eef066d2893edbd828b0e9d7d1d55a.png[/IMG].
Sorry for the large text (I don’t know how that happened, something wrong with the copy and paste).
“You should check out the forums on AoPS if you want to find students like that. They’re generally not here.
”
Exactly. And this recommendation insight is for the students who are here.
“High school kids knowing abstract algebra? I’m not saying that it doesn’t happen, but I’ve honestly never talked to such a student before. And I’ve talked to hundreds of students before on PF.”
You should check out the forums on AoPS if you want to find students like that. They’re generally not here.
“Also, abstract algebra is a huge field. It takes years before you really get the basics of it. Do you really mean to say that high school students know the Sylow theorems? The fundamental theorem of Galois correspondence? The Nullstellensatz? The chinese remainder theorem? etc. Somehow I find this very very very very hard to believe…”
Here’s the Art of Problem Solving [URL=’http://www.artofproblemsolving.com/school/course/catalog/grouptheory’]course on Group Theory[/URL], which has been around for only a couple of years. So you can be assured there are quite a few high school kids learning at least to this level. I’m sure some go deeper.
My kid first started studying on his own out of Herstein’s [URL=’http://www.amazon.com/Topics-Algebra-2nd-Edition-Herstein/dp/0471010901′]Topics in Algebra[/URL] when he was 13 (he got through only a couple of chapters). Later he used Jacobson’s [URL=’http://www.amazon.com/Basic-Algebra-Second-Dover-Mathematics/dp/0486471896′]Basic Algebra I[/URL] & [URL=’http://www.amazon.com/Basic-Algebra-II-Second-Mathematics/dp/048647187X’]II[/URL]. By the time he went off to college he had spent five years studying modern algebra with varying degrees of intensity. One of the last algebra related things he did before going off to Cambridge was a graduate course on algebraic number theory followed by some readings on ramification from Serre’s [URL=’http://www.amazon.com/Local-Fields-Graduate-Texts-Mathematics/dp/0387904247′]Local Fields[/URL]. His experience is very uncommon, but not unknown. It’s going to be very interesting to see what the level of the kids coming out of [URL=’https://en.wikipedia.org/wiki/Proof_School’]Proof School[/URL] will be in a couple of years.
High school kids knowing abstract algebra? I’m not saying that it doesn’t happen, but I’ve honestly never talked to such a student before. And I’ve talked to hundreds of students before on PF. acegikmoqsuwy, you’re clearly too advanced to get something useful out of this guide. But it’s not fair to say there are many high school students with a good grasp on projective geometry and abstract algebra. I really don’t know where you get this from.
Also, abstract algebra is a huge field. It takes years before you really get the basics of it. Do you really mean to say that high school students know the Sylow theorems? The fundamental theorem of Galois correspondence? The Nullstellensatz? The chinese remainder theorem? etc. Somehow I find this very very very very hard to believe…
”
Anyway, in my opinion, I agree that the article seems “outdated” in a sense, since the majority of high schoolers I know that are interested math already know topics including Projective Geometry, Graph Theory, and Abstract Algebra and are able to use them with moderate success on proof contests. The biggest problem with higher level math that’s come about in my high school, as well as for many other high schoolers I know, is that even the local college courses are not enough; some kids complete Calculus I-III, Linear Algebra, and Differential Equations by their 9th grade year and then are stuck because they’ve exhausted all the classes from their community colleges and cannot afford the higher level classes at the actual colleges (since the school only pays a small fraction of the cost). As for me, I was slowed a tad since I was only placed in Geometry in 8th grade, but even as I have worked up to Differential Equations in 10th grade, I find myself stuck in the exact same issue.
”
(Bold added by me for emphasis)
Where are you going to school that allows you to proceed so quickly to college-level math? :wideeyed:
It’s not that I don’t believe you or that I mean to demean your frustrations, but my experience is not at all similar to yours (I’m currently a sophomore in high school, by the way).
The majority of high school students that I know that are “interested in math” (not quite sure what you mean by that – do you mean people who enjoy math or people who are seriously thinking about math/physics as a career?) do not know Projective Geometry, Graph Theory, and Abstract Algebra. Although, your latter statement about proof contests makes me think that you and I have different definitions of “know.” Being involved in contest math myself, one does not necessarily have to understand the topic fully to get the right answer. So do they actually understand the ins and outs of it, or are they just comfortable with the basics (neither of which are bad, but that part of your post seems a bit ambiguous to me)?
In addition, I have never met anyone who has completed Calculus I-III, Linear Algebra, and Diff EQ by their 9th grade year. Is this the norm for the very advanced students in your area?
Also, with your statement:
”
I was slowed a tad since I was only placed in Geometry in 8th grade, but even as I have worked up to Differential Equations in 10th grade, I find myself stuck in the exact same issue.
”
I am a bit in awe since I have no opportunity to attend a Diff EQ class right now (as a sophomore) and being in Geometry in 9th grade is the highest you can normally do in my school (and other schools around my area).
So, in my opinion, micromass’ post does not seem meant to the student of forty years ago (unless my area is just full of bums who are complete underachievers – which is possible but unlikely). He gives excellent suggestions, and as someone currently trying to do some advanced mathematics, I can say that I find the directions in which he points to be very helpful. The only reason why I didn’t get more out of the Insight is because I’ve spent some time on these forums and seen his other posts where he mentions similar things and already started on one of the areas he mentions (namely, linear algebra – and I see he’s got a recent post on that, too!).
EDIT: Sorry if weirdness happened with the quoted section of my comment on the actual Insight page. I posted it in the regular forums and for some reason the quote brackets aren’t showing up on the other page.
“… some kids complete Calculus I-III, Linear Algebra, and Differential Equations by their 9th grade year and then are stuck because they’ve exhausted all the classes from their community colleges and cannot afford the higher level classes at the actual colleges (since the school only pays a small fraction of the cost).”
The solution to that problem for my math kid was to just ignore getting credit for anything. He audited a large variety of classes at local universities while in middle school and high school. Never officially, always by just asking permission of the professor who taught the class. Mostly grad classes after the first couple. It worked wonderfully and cost nothing (except for books). However I did pull him out of school to homeschool him, which made it possible to attend college classes in the middle of the day — that’s difficult to do if you are still going to high school.
He also learned things on his own. For calculus he worked his was through Apostol I & II, doing all the problems (not all the practice exercises though). Similarly he self-studied various other topics from the best books I could find for him (e.g. he learned real analysis from Tao’s notes for [URL=’https://terrytao.files.wordpress.com/2012/12/gsm-117-tao3-epsilon1.pdf’]An Epsilon of Room[/URL]).
“Another thing I would like to point out: in my opinion, the issue with all the high school and college classes is not that the material is easy (since the material being taught is not going to change no matter how “hard” you make the class), but rather in the problems given. Most problems that I have seen consist of trivial observations from the definition, or “plug-and-chug.””
Yeah, it’s pretty much impossible to get courses meant for mathematicians at community colleges — it’s not their audience. Actually it’s hard at most universities (just count the ones using Stewart for calculus rather than Apostol). So with pretty much everything you have to take responsibility upon yourself for learning the material properly. Good training for the rest of life.
“This is what makes (in my opinion), mathematics competitions far more interesting; you are not supposed to know how to solve the problem, rather you are supposed to figure out on your own what the key ingredients are that are needed to solve the problem. That is the art of problem solving.”
Yeah, that’s the good part. But the problem with competition problems is that they are known to be solvable in a pretty short time — this is completely unlike the sorts of problems mathematicians work on. And you have to be a fast thinker to do them, which discourages the slow, deep thinkers who are the ones most likely to succeed as research mathematicians. So it’s a mixed bag. Works for some people, not others.
“Good stuff, but it might as well have been written for the student of forty years ago. Nowadays the world is full of students in high school doing advanced mathematics. And most of them use materials from [URL=’http://www.artofproblemsolving.com’]Art of Problem Solving[/URL]. Their books are excellent and fairly priced. Their classes are also excellent, but kind of expensive and not suitable for all. But for kids who love math and are good at it there’s nothing better out there. Also check out their [URL=’http://www.artofproblemsolving.com/alcumus’]Alcumus[/URL] online problem system. Free and useful. They also support social interaction and problem solving with online forums, also free and useful.”
I agree one hundred percent. I am currently a high schooler and AoPS is my regular go-to for anything to do with math, whether it be studying for competitions, learning higher level math, or perhaps if I’m just bored and want someone to talk to about math. The part which really caught my eye the first time I ever went on AoPS was the following quote on their page: “Is math class too easy for you? You’ve come to the right place!”
Anyway, in my opinion, I agree that the article seems “outdated” in a sense, since the majority of high schoolers I know that are interested math already know topics including Projective Geometry, Graph Theory, and Abstract Algebra and are able to use them with moderate success on proof contests. The biggest problem with higher level math that’s come about in my high school, as well as for many other high schoolers I know, is that even the local college courses are not enough; some kids complete Calculus I-III, Linear Algebra, and Differential Equations by their 9th grade year and then are stuck because they’ve exhausted all the classes from their community colleges and cannot afford the higher level classes at the actual colleges (since the school only pays a small fraction of the cost). As for me, I was slowed a tad since I was only placed in Geometry in 8th grade, but even as I have worked up to Differential Equations in 10th grade, I find myself stuck in the exact same issue.
Another thing I would like to point out: in my opinion, the issue with all the high school and college classes is not that the material is easy (since the material being taught is not going to change no matter how “hard” you make the class), but rather in the problems given. Most problems that I have seen consist of trivial observations from the definition, or “plug-and-chug.” This is what makes (in my opinion), mathematics competitions far more interesting; you are not supposed to know how to solve the problem, rather you are supposed to figure out on your own what the key ingredients are that are needed to solve the problem. That is the art of problem solving.
thanks alot
could you please offer some books about Lie algebra (An, Bn,…)and exceptional like G2 algebra in physics with easy language?
thanks again
“micromass submitted a new PF Insights post
[URL=’https://www.physicsforums.com/insights/high-school-want-advanced-mathematics/’]In High School and Want to Do Advanced Mathematics?[/URL]…”
Good stuff, but it might as well have been written for the student of forty years ago. Nowadays the world is full of students in high school doing advanced mathematics. And most of them use materials from [URL=’http://www.artofproblemsolving.com’]Art of Problem Solving[/URL]. Their books are excellent and fairly priced. Their classes are also excellent, but kind of expensive and not suitable for all. But for kids who love math and are good at it there’s nothing better out there. Also check out their [URL=’http://www.artofproblemsolving.com/alcumus’]Alcumus[/URL] online problem system. Free and useful. They also support social interaction and problem solving with online forums, also free and useful.
My math kid used several of their books a number of years ago, both when he was in public school and needed more and better materials, and later when he was home schooled. It was a revelation. AoPS was started by a bunch of math guys who set out to create the things they wished were available when they were in high school. I think they’ve succeeded admirably. You would be doing yourself a favor to become familiar with it.
Wow. This insight is awesome. Thanks a lot micromass.
“Great post.
I have a question : How does one gain intuition about matrices, determinants? I can find inverses and solve determinants using properties but I don’t think I truly understand them. Does linear algebra provide an insight into them?”
You gain intuition about matrices and determinants only by studying the underlying geometry. This geometry is very naturally that of vector spaces and linear transformations. A matrix is then simply a very easy way to represent linear transformations, and the determinant is simply how the linear transformation acts on the volume of the unit cube. The annoying point is that it is very recommended to be able to compute with matrices and determinants before you should handle vector spaces. The effect is then that you compute with matrices without seeing what they really are. You’ll have to get through this, I fear.
“Great post.
I have a question : How does one gain intuition about matrices, determinants? I can find inverses and solve determinants using properties but I don’t think I truly understand them. Does linear algebra provide an insight into them?[/quote]That IS linear algebra, get a good book and work through it and see for yourself. I assume you just learned how to do it in your algebra class and the teacher didn’t explain much. If so, then I was in the same situation until I took linear algebra.[/quote]Sorry strange format
“Great post.
I have a question : How does one gain intuition about matrices, determinants? I can find inverses and solve determinants using properties but I don’t think I truly understand them. Does linear algebra provide an insight into them?[/quote]That IS linear algebra, get a good book and work through it and see for yourself. I assume you just learned how to do it in your algebra class and the teacher didn’t explain much. If so, then I was in the same situation until I took linear algebra.
Great post.
I have a question : How does one gain intuition about matrices, determinants? I can find inverses and solve determinants using properties but I don’t think I truly understand them. Does linear algebra provide an insight into them?
Micromass, this insight was written like it was just for me. Thanks so much . . . :smile:
thanks alotcould you please offer some books about Lie algebra (An, Bn,…)and exceptional like G2 algebra in physics with easy language?thanks again
Wow. This insight is awesome. Thanks a lot micromass.
Sorry strange format
That IS linear algebra, get a good book and work through it and see for yourself. I assume you just learned how to do it in your algebra class and the teacher didn't explain much. If so, then I was in the same situation until I took linear algebra.
Great post. I have a question : How does one gain intuition about matrices, determinants? I can find inverses and solve determinants using properties but I don't think I truly understand them. Does linear algebra provide an insight into them?