University Math for High Schoolers: Where to Start
High school students who want a taste of university-level mathematics before they graduate can start now with subjects like abstract algebra, linear algebra, Euclidean geometry, and non-Euclidean geometries. These topics require few or no calculus prerequisites, and accessible textbooks exist for each. High school coursework should still be completed in full, since it remains the foundation for everything that follows.
Table of Contents
Key Takeaways
- Abstract algebra can be started with no calculus background, using a book such as Pinter’s “A Book of Abstract Algebra.”
- Linear algebra requires comfort with vectors, the dot product, and Gaussian elimination, and can be studied through Alan Macdonald’s “Linear and Geometric Algebra.”
- Euclid’s “Elements” has no specific prerequisites beyond motivation, and working through its first four books leads to constructions such as the regular pentagon.
- Affine, projective, spherical, and hyperbolic geometry require prior knowledge of Euclidean geometry, matrix computations, and basic group theory.
- Mathematics competitions are a separate activity from university mathematics, and success in one does not require success in the other.
Should high school students skip ahead to university mathematics?
High school students who feel unchallenged by their courses do not need to wait until university to encounter serious mathematics. Some university-level topics require heavy prerequisites, such as multiple semesters of calculus, and those aren’t accessible yet. Many other topics, however, can be started with little more than a solid grip on high school algebra and geometry.
Exploring university mathematics early does not mean high school coursework can be skipped. Skills like factoring polynomials are unglamorous but necessary, and neglecting them creates problems later. Being able to work through an abstract algebra textbook does not substitute for mastering the material a high school course requires.
Do you need to enjoy math competitions to enjoy university mathematics?
Mathematics competitions can be enjoyable for students who like that format, but they are not a requirement for enjoying or succeeding at university-level mathematics. I personally never enjoyed competitions and failed every one I entered. Competitions test a different skill set than the one used in university mathematics courses, and doing poorly at one says little about ability in the other.
What is abstract algebra and what does it require?
Abstract algebra typically does not require calculus or trigonometry. Any examples that use those tools can usually be skipped without losing the thread of the subject. Pinter’s “A Book of Abstract Algebra” is a solid introductory text, and Armstrong’s “Groups and Symmetry” offers a more geometric approach to the same material.
Prerequisites: solving polynomial equations including the quadratic formula, basic set theory (sets, functions, unions, intersections, Cartesian products), elementary number theory (prime numbers and the fundamental theorem of arithmetic), basic proof methods such as mathematical induction and proof by contradiction, and comfort with vectors and equations of lines and planes. Familiarity with ruler-and-compass constructions is helpful but not required.
What you’ll learn: group theory, which describes symmetries ranging from the symmetries of a cube to symmetries in physical theories; ring theory, which has applications in coding theory and algebraic geometry and deepens the study of the fundamental theorem of arithmetic; and field theory, which explores why most polynomial equations have no solution expressible by a simple formula.
What is linear algebra and what does it require?
Linear algebra extends geometry into arbitrary dimensions, showing what points, lines, and planes look like in high-dimensional spaces. Many of its methods are central to science and engineering, making it one of the more directly applicable subjects on this list. MacDonald’s “Linear and Geometric Algebra” is recommended as an accessible introduction to geometric algebra.
Prerequisites: comfort with vectors, vector addition, and the dot product from geometry; familiarity with matrices, including solving linear equations by Gaussian elimination, matrix multiplication, and matrix inversion. Basic proof technique and set theory are helpful but not essential.
What you’ll learn: the abstract notion of a vector space, a concept that appears throughout physics and engineering; geometric algebra, a useful but often overlooked framework for geometric computation; and the theory of linear maps and matrix diagonalization, both widely used tools.
Why study Euclid’s Elements today?
Euclid’s “Elements” remains one of the most important mathematics books ever written, and it has no specific prerequisite beyond motivation and willingness to work through the material. The full text is available online through Clark University’s archive, and Casey and Callahan’s “Euclid’s Elements Redux” offers a modern edition for readers who prefer updated notation. Hartshorne’s “Geometry: Euclid and Beyond” provides advanced commentary, though its later chapters assume abstract algebra.
Working through at least the first four books of Euclid develops geometry from first principles and culminates in classical constructions such as the regular pentagon, an elegant and instructive result in its own right.
What are affine, projective, spherical, and hyperbolic geometry?
These geometries extend beyond the Euclidean plane into settings that were decisive for my own interest in mathematics when I first encountered them in high school. Brannan, Esplen, and Gray’s “Geometry” covers this material and requires somewhat more background than typical high school courses, though the gap is bridgeable with focused preparation.
Prerequisites: solid knowledge of Euclidean geometry including vector computation and equations of lines and planes; matrix skills covering multiplication, inversion, diagonalization, and solving linear systems (introductory videos such as those from Khan Academy can cover diagonalization if it hasn’t been seen before); algebra skills for solving polynomial equations; and some basic group theory, for which the opening chapters of Pinter or Armstrong are sufficient.
What you’ll learn: conic sections and their applications, including practical properties of parabolas; projective geometry, which has applications in image processing; spherical geometry and its many applications; and hyperbolic geometry, a strikingly different geometry with important connections to relativity.
What other subjects are worth exploring?
John Stillwell’s books, including “The Four Pillars of Geometry” and “Roads to Infinity,” are worth an honorable mention for students looking beyond the four subjects above. Stillwell is regarded as an excellent expositor, and his range of topics means most students will find something that appeals to them.
Frequently Asked Questions
Do I need calculus before starting abstract algebra or linear algebra?
No. Abstract algebra typically does not require calculus or trigonometry, and any examples that touch on those subjects can usually be skipped. Linear algebra requires comfort with vectors and matrices rather than calculus, so both subjects are accessible to students who haven’t yet taken calculus courses.
Should I stop doing my regular high school math homework to focus on these topics?
No. High school coursework, including repetitive tasks like factoring polynomials, builds skills that remain necessary and that you will need later. Being able to work through university-level material does not mean high school math can be skipped.
Do I need to be good at math competitions to study these topics?
No. Math competitions test a different skill set than university mathematics, and it is entirely possible to succeed at the latter without enjoying or excelling at the former.
Which book should I start with if I have never seen a formal proof?
Euclid’s “Elements” requires no specific prerequisites beyond motivation and is a natural starting point for students unfamiliar with formal proof, since it develops geometry step by step from first principles.
What comes after abstract algebra and linear algebra?
Affine, projective, spherical, and hyperbolic geometry build on knowledge of Euclidean geometry, matrix computation, and basic group theory, making them a reasonable next step once the earlier subjects have been covered.
Are these subjects useful outside of pure mathematics?
Yes. Linear algebra methods are central to science and engineering, projective geometry has applications in image processing, and hyperbolic geometry has important connections to relativity.
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?