On Linear Algebra

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Phi
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On Linear Algebra

Postby Phi » Thu Nov 08, 2007 7:22 am UTC

To those who are/have taken Linear Algebra, how much background Calculus knowledge needs to be known, if any? I have talked to my math teacher right now (Currently in Calculus BC, not exactly sure how that translates to I, II, etc. (but if you have Calculus I-IV that's Calculus II & III, if that's any help), and he claims that you really don't need to know any Calculus, unless you are taking the Engineering form of it, in which case you only really need to know systems of differential equations.

Here's why I'm curious: my friend is taking courses over at the local community college instead of staying at the high school taking courses there. He calls me up one day and asks if I'd be interested in taking Linear Algebra over the summer, after a year of Calculus. I said yes and started to look up requirements. Calculus D (IV), Multi-variable Calculus. Do I need to know multi-variable calculus in order to perform in Linear Algebra? Again, my math teacher says that I do not, and I am inclined to believe him.

Now here's the real kicker. Proving that you got a 5 on the BC AP exam only exempts you from taking Calculus A and B, but not C. Which means that I'd have to take C again, as a repeat instead of being able to continue on to Calculus D. (All other colleges I know of accept that 5 as an exemption from A-C, but not this one community college, no) And all of this just to be able to take Linear Algebra.

So this is my question, re-stated: How much am I missing out on? Am I getting screwed over by the system, so to speak? And is there a way to skip out on the course in college because I somehow "already know it" (i.e. grab my friend's textbook and start reading)?

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Re: On Linear Algebra

Postby btilly » Thu Nov 08, 2007 7:43 am UTC

Phi wrote:To those who are/have taken Linear Algebra, how much background Calculus knowledge needs to be known, if any? I have talked to my math teacher right now (Currently in Calculus BC, not exactly sure how that translates to I, II, etc. (but if you have Calculus I-IV that's Calculus II & III, if that's any help), and he claims that you really don't need to know any Calculus, unless you are taking the Engineering form of it, in which case you only really need to know systems of differential equations.


Your teacher is right. Linear algebra does not involve Calculus at all. There are lots of applications of linear algebra which also require you to know Calculus, but the subject itself does not.

As for what it is, the basic idea of linear algebra is that in many cases you want to organize numbers into rectangular matrices and manipulate those in various ways. And by so organizing a bunch of numbers, you can model all sorts of interesting things.

If this idea seems strange to you, consider solving 3 equations in 3 variables. Or 4 equations in 4 variables. Or 5 equations in 5 variables. What purpose do the variable names have when you do that? None! You could state the same problem, and the same solutions, more compactly if you forgot your variables and arranged the coefficients into a rectangle and began adding and subtracting multiples of one row to another. When you're done, then you can remember the variable names.

(Many introductory linear algebra courses start with this exact problem.)

So this is my question, re-stated: How much am I missing out on? Am I getting screwed over by the system, so to speak? And is there a way to skip out on the course in college because I somehow "already know it" (i.e. grab my friend's textbook and start reading)?


At many colleges you always have the option of challenging a course that you feel you already understood. That just means that you take the final exam. If you pass, then they'll accept that you've taken the material and let you go to more advanced material.

Furthermore in many cases you can get transfer credit from a community college to wherever you go to college. So the work done may apply.

(The qualifiers are because colleges have a lot of latitude to make their own rules up, so no hard and fast rule will apply to all of them.)
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Re: On Linear Algebra

Postby Buttons » Thu Nov 08, 2007 7:53 am UTC

Phi wrote:To those who are/have taken Linear Algebra, how much background Calculus knowledge needs to be known, if any? I have talked to my math teacher right now (Currently in Calculus BC, not exactly sure how that translates to I, II, etc. (but if you have Calculus I-IV that's Calculus II & III, if that's any help), and he claims that you really don't need to know any Calculus, unless you are taking the Engineering form of it, in which case you only really need to know systems of differential equations.

He's correct. Though linear algebra curricula vary from school to school (and professor to professor), there's little to no multivariable calculus involved. You'll frequently see some basic Calc I stuff (for instance, linear transformations involving derivatives are polynomials are fairly common in homework problems), but nothing too advanced. One professor here covers Jacobians, but otherwise I can't really think of any reason you'd need to take multivariable calculus to understand the material.

If such material isn't used, then why is it a prerequisite, you ask? I think most schools arrange their introductory math sequence so as to build mathematical maturity. Multivariable calculus comes after single-variable calculus not because you need to understand Taylor series in order to do Lagrange multipliers, but because it's a more conceptual class that requires a lot of comfortability with the ideas of calculus. Linear algebra is even more conceptual (though, again, this depends on the professor), and it's often the first class people take where they regularly have to prove things on homework problems. It's offered after the calculus sequence because of its abstractness, not because of its content.
Phi wrote:Now here's the real kicker. Proving that you got a 5 on the BC AP exam only exempts you from taking Calculus A and B, but not C. Which means that I'd have to take C again, as a repeat instead of being able to continue on to Calculus D. (All other colleges I know of accept that 5 as an exemption from A-C, but not this one community college, no) And all of this just to be able to take Linear Algebra.
Professors don't want to teach students who already know the material any more than those students want to waste a semester in a course they've already taken. If you really don't think you'd be learning anything in Calc C, talk to the professor, and I suspect they'll let you skip ahead. The official position of requiring X score to place into Y course is often stricter than necessary, so as to prevent people from accidentally signing up for courses they aren't ready for. If you don't think the standard applies, you're probably right.

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Re: On Linear Algebra

Postby Phi » Thu Nov 08, 2007 8:12 am UTC

btilly wrote:If this idea seems strange to you, consider solving 3 equations in 3 variables. Or 4 equations in 4 variables. Or 5 equations in 5 variables. What purpose do the variable names have when you do that? None! You could state the same problem, and the same solutions, more compactly if you forgot your variables and arranged the coefficients into a rectangle and began adding and subtracting multiples of one row to another. When you're done, then you can remember the variable names.

Correct me if I'm wrong, but isn't this just a system of equations in a matrix? Reduced Row Echelon Form, etc? What else do I get into? What purpose does Linear Algebra serve in the real world?
btilly wrote:At many colleges you always have the option of challenging a course that you feel you already understood. That just means that you take the final exam. If you pass, then they'll accept that you've taken the material and let you go to more advanced material.

Ah, yes, I forgot about this option. I really do think I will be doing this if my friend does end up taking his Linear Algebra course over the summer.

Buttons wrote:...it's often the first class people take where they regularly have to prove things on homework problems. It's offered after the calculus sequence because of its abstractness, not because of its content.

Shouldn't you regularly be proving things to yourself during homework problems anyway? Isn't that the point of your education?
If you aren't "mathematically mature" enough to handle it before Calculus, and the Calculus class is handled in this way, I don't see how you'd be any more ready for it later on, especially if it's still more abstract.

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Re: On Linear Algebra

Postby scarletmanuka » Thu Nov 08, 2007 8:25 am UTC

I'd think it would make more sense to do Linear Algebra before multivariate calculus, because parts of multivariate calculus require some linear algebra (for instance, integration with coordinate transformations).

Anyway, after the introductory material on systems of linear equations, the next step is looking at linear transformations, which can also be expressed in terms of matrices. The steps used to solve systems of linear equations by applying certain elementary procedures turn out to be equivalent to decomposing an arbitrary linear transformation into a combination of elementary transformations. There's a bunch of stuff on vector spaces and their properties, really quite a lot of interesting things at this level. Lots of applications in three-dimensional geometry (most of which generalises in some way to higher dimensions too).

Linear algebra is really a foundational skill that has applications in many areas of mathematics. Once you get into it, it starts cropping up all over the place. I'd definitely recommend that you find some way to do it!

P.S. Here in Western Australia basic linear algebra is taught in the final year of high school (at the time I studied it, this was in the first-year syllabus at university; it's since been brought into the high school syllabus). Multivariate calculus isn't until a couple of years later. :)

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Re: On Linear Algebra

Postby btilly » Thu Nov 08, 2007 9:44 am UTC

Phi wrote:
btilly wrote:If this idea seems strange to you, consider solving 3 equations in 3 variables. Or 4 equations in 4 variables. Or 5 equations in 5 variables. What purpose do the variable names have when you do that? None! You could state the same problem, and the same solutions, more compactly if you forgot your variables and arranged the coefficients into a rectangle and began adding and subtracting multiples of one row to another. When you're done, then you can remember the variable names.

Correct me if I'm wrong, but isn't this just a system of equations in a matrix? Reduced Row Echelon Form, etc? What else do I get into? What purpose does Linear Algebra serve in the real world?


You're right, and that is the start of linear algebra.

In the real world, linear algebra shows up in many ways. First of all if you have to describe complicated systems with many variables that feed back into each other in an approximately linear way (eg an economy, or a complicated electrical circuit), linear algebra provides the language you use to do it.

Secondly if you have a function from many variables to many variables, you need a higher-dimensional analog of "the tangent line". Linear algebra provides the language to be able to describe that. This makes linear algebra very important to multi-variable calculus.

Thirdly there are a lot of abstract mathematical systems that arise from linear algebra which have proven important. (For instance they show up in places like quantum mechanics.)

And so on and so forth.

Plus some people find the subject interesting in its own right. :)
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Re: On Linear Algebra

Postby dosboot » Thu Nov 08, 2007 9:46 am UTC

Phi wrote:Correct me if I'm wrong, but isn't this just a system of equations in a matrix? Reduced Row Echelon Form, etc? What else do I get into? What purpose does Linear Algebra serve in the real world?


I've never taken a pure linear algebra course, but I'm fairly sure there isn't a whole lot to linear algebra besides solving systems of equations if the course doesn't dabble into multivariable calculus. The only other non-calculus topic that comes to mind is linear dependence/independence of vectors, which is a linear equation problem in disguise. Solving systems of linear equations however is very important in math and in the real world.

If it does go into multivariable calculus, then expect topics like: the Jacobian matrix, tangent (hyper)planes, Lagrange multipliers, change of variables for multivariate integration.

Phi wrote:Shouldn't you regularly be proving things to yourself during homework problems anyway? Isn't that the point of your education? If you aren't "mathematically mature" enough to handle it before Calculus, and the Calculus class is handled in this way, I don't see how you'd be any more ready for it later on, especially if it's still more abstract.


In almost all colleges, some amount of Calculus is required no matter the major. The non-science/non-math majors will take a completely non-proof based Calculus course since they aren't ever going to take another math course again. It is not done at the level of what most people see in AP calculus either. Then there is usually a special 'hard' Calculus sequence for the science/math majors to take.

Let's cut to the chase here: are you good at math? It sounds like you've already seen stuff from both Calculus C and Linear algebra. It is possible that one or both of those courses are not going to teach anything. This isn't all that uncommon, what you do in these situations is talk to professors in the math department. If you are just trying to get into a specific class without having the proper requirements then you just talk to whoever is teaching the class. If you are trying to get 'official' credit for a class on your transcript you'll probably have to talk to the math department head too and see what's up. In general if you are a math major you should have conversations with your advisor about what courses you should be taking and what courses you should _not_ be taking (i.e. those that would be below you).

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Re: On Linear Algebra

Postby rhino » Thu Nov 08, 2007 2:18 pm UTC

dosboot wrote:I've never taken a pure linear algebra course, but I'm fairly sure there isn't a whole lot to linear algebra besides solving systems of equations if the course doesn't dabble into multivariable calculus. The only other non-calculus topic that comes to mind is linear dependence/independence of vectors, which is a linear equation problem in disguise. Solving systems of linear equations however is very important in math and in the real world.

Well... that is just totally untrue! Saying that linear algebra is just solving systems of equations is like saying that number theory is just counting.

Pure linear algebra is the study of general vector spaces and linear maps. These are fundamental mathematical objects and their theory has applications all over the place, both in the real world and other areas of pure maths. To name a few examples: quantum mechanics, stochastic processes, functional analysis, representation theory, numerical analysis and vast areas of calculus.

I 100% recommend taking it if you have any interest in pure maths, but even if you don't, it will make your life much easier when you come to study subsequent topics in applied maths. And no, it really isn't just solving linear systems of equations by Gaussian elimination.

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Re: On Linear Algebra

Postby btilly » Thu Nov 08, 2007 4:35 pm UTC

dosboot wrote:
Phi wrote:Correct me if I'm wrong, but isn't this just a system of equations in a matrix? Reduced Row Echelon Form, etc? What else do I get into? What purpose does Linear Algebra serve in the real world?


I've never taken a pure linear algebra course, but I'm fairly sure there isn't a whole lot to linear algebra besides solving systems of equations if the course doesn't dabble into multivariable calculus. The only other non-calculus topic that comes to mind is linear dependence/independence of vectors, which is a linear equation problem in disguise. Solving systems of linear equations however is very important in math and in the real world.


Translation, "I'm an idiot and I'm willing to let the whole world know about it."

Sorry to be so harsh about it, but why would you say such a thing? Particularly when you know you don't know much about the subject, and there are already replies indicating that there is more to the topic than you know?

If you want a sample of some linear algebra that you haven't seen, look at some of the structure theorems. There is a nice presentation of them from scratch at http://www.axler.net/DwD.html. If you can make it through that paper, you'll have learned something about linear algebra.
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Re: On Linear Algebra

Postby Torn Apart By Dingos » Thu Nov 08, 2007 4:56 pm UTC

You don't need to know any calculus. Linear algebra is great. It's useful in pretty much everything. One thing that's good about it is that it uses abstract structures like vector spaces and inner products that have a very intuitive interpretation, but because they're abstract, you can apply the theory to things that aren't points or planes. For example, the Gram-Schmidt process is something which is very obvious when you apply it to (geometric) vectors, but using an integral as an inner product for functions, you can use the same theory to project functions onto sets of functions (ie get the linear combination of the set of functions closest to the function you started with), which isn't at all as obvious how to do it.

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Re: On Linear Algebra

Postby Zohar » Thu Nov 08, 2007 6:20 pm UTC

I studies math and physics at an undergrad level and IMO, rigorous knowledge of linear algebra (especially anything about eigenvalues) is much more important than a rigorous knowledge of calculus. Calculus at the basic level you use in undergrad physics studies is fairly intuitive. Linear algebra is immensely important.
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Re: On Linear Algebra

Postby Yakk » Thu Nov 08, 2007 6:52 pm UTC

btilly wrote:As for what it is, the basic idea of linear algebra is that in many cases you want to organize numbers into rectangular matrices and manipulate those in various ways. And by so organizing a bunch of numbers, you can model all sorts of interesting things.

If this idea seems strange to you, consider solving 3 equations in 3 variables. Or 4 equations in 4 variables. Or 5 equations in 5 variables. What purpose do the variable names have when you do that? None! You could state the same problem, and the same solutions, more compactly if you forgot your variables and arranged the coefficients into a rectangle and began adding and subtracting multiples of one row to another. When you're done, then you can remember the variable names.


That's linear programming or linear optimization moreso than linear algebra.

Note that if the course requires calc 3, then it is quite possible that the course rather than the subject material relies on concepts taught in the calc 3 course.


Phi wrote:Correct me if I'm wrong, but isn't this just a system of equations in a matrix? Reduced Row Echelon Form, etc? What else do I get into? What purpose does Linear Algebra serve in the real world?


A matrix is one way of representing a linear transformation in a finite dimensional vector space.

Linear algebra is the study of Vector Spaces, usually over the Reals. It deals with the concept of Linear Transformation, Metrics, Norms, Inner Products, Outer Products, Duals, Kernels, Dimension, Determinates, etc. You can go on from there to Infinite dimensional vector spaces, vector spaces over alternative fields (such as Z/2), the Ln metrics, and eventually up to things like spectral theory. Other branches wander off into the realms of relativity, quantum mathemtics, and network optimization.

Oh, and for the application junkies around here, Affine Algebra is at the core of most 3D graphics nowadays, and Affine Algebra is a variant of Linear Algebra.
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Re: On Linear Algebra

Postby jtniehof » Thu Nov 08, 2007 7:52 pm UTC

Piggy-backing:
I seem to have lost my linear algebra text, which I thought was darn good and would like to buy another copy. Does a white text with a quilt-like pattern on the cover, fairly small form factor, ring a bell for anyone? It'd be published before 1999 at the latest. (The title is completely nondescript.)

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Re: On Linear Algebra

Postby btilly » Thu Nov 08, 2007 8:25 pm UTC

Yakk wrote:
btilly wrote:As for what it is, the basic idea of linear algebra is that in many cases you want to organize numbers into rectangular matrices and manipulate those in various ways. And by so organizing a bunch of numbers, you can model all sorts of interesting things.

If this idea seems strange to you, consider solving 3 equations in 3 variables. Or 4 equations in 4 variables. Or 5 equations in 5 variables. What purpose do the variable names have when you do that? None! You could state the same problem, and the same solutions, more compactly if you forgot your variables and arranged the coefficients into a rectangle and began adding and subtracting multiples of one row to another. When you're done, then you can remember the variable names.


That's linear programming or linear optimization moreso than linear algebra.


It also shows up in introductory linear algebra courses. Generally near the beginning.

Yakk wrote:Note that if the course requires calc 3, then it is quite possible that the course rather than the subject material relies on concepts taught in the calc 3 course.


Phi wrote:Correct me if I'm wrong, but isn't this just a system of equations in a matrix? Reduced Row Echelon Form, etc? What else do I get into? What purpose does Linear Algebra serve in the real world?


A matrix is one way of representing a linear transformation in a finite dimensional vector space.

Linear algebra is the study of Vector Spaces, usually over the Reals. It deals with the concept of Linear Transformation, Metrics, Norms, Inner Products, Outer Products, Duals, Kernels, Dimension, Determinates, etc. You can go on from there to Infinite dimensional vector spaces, vector spaces over alternative fields (such as Z/2), the Ln metrics, and eventually up to things like spectral theory. Other branches wander off into the realms of relativity, quantum mathemtics, and network optimization.


Of course as you go along you can get more abstract. But I was trying to keep the discussion to a level that a novice would understand.

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Re: On Linear Algebra

Postby Yakk » Thu Nov 08, 2007 8:37 pm UTC

Ah yes, from that end...

A vector is a quantity with a direction, and no other properties.

You can scale a vector (double it, half it, multiply it by pi). This produces a vector.
You can add two vectors together, and produce a vector.

There are a special set of functions, known as "linear", that behave nicely with scaling and adding of vectors.

Linear Algebra is the study of Vectors and Linear Transformations on them. It is the study of Spaces of Vectors (an infinite sheet of paper might be a space in which your "directions" are defined).

This gives you useful things. Right off the bat, it makes solving systems of equations much easier. You learn neat ways to express things like rotations and rescales, which is the basis of lots of computer graphics. The Linear Transformation turns out to be a really close analog to the concept of "slope", which leads back into calculus.

And on it's back, you can climb up to dizzying heights. Climb onto the shoulders of a giant, and look in awe at the mountain to come. :)
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Re: On Linear Algebra

Postby Token » Thu Nov 08, 2007 9:38 pm UTC

Yakk wrote:A vector is a quantity with a direction, and no other properties.

Well, strictly speaking, a vector is any element of a vector space. It isn't required to have direction, and can certainly have other properties - for example, "converging to zero" seems like a reasonable non-direction property for an element of the space of real-valued sequences.
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Re: On Linear Algebra

Postby Herman » Thu Nov 08, 2007 9:50 pm UTC

jtniehof wrote:Piggy-backing:
I seem to have lost my linear algebra text, which I thought was darn good and would like to buy another copy. Does a white text with a quilt-like pattern on the cover, fairly small form factor, ring a bell for anyone? It'd be published before 1999 at the latest. (The title is completely nondescript.)


That sounds like the book I'm using in the LA class I have right now. (Weeeirrd...)

Linear Algebra and its Applications
Gilbert Strang
4th Ed.
ISBN 0030105676

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Re: On Linear Algebra

Postby Yakk » Thu Nov 08, 2007 9:54 pm UTC

Token wrote:
Yakk wrote:A vector is a quantity with a direction, and no other properties.

Well, strictly speaking, a vector is any element of a vector space. It isn't required to have direction, and can certainly have other properties - for example, "converging to zero" seems like a reasonable non-direction property for an element of the space of real-valued sequences.


:p Yes, I'm aware. But I was attempting to generate a "basic walkthrough" of what Linear Algebra was about. I already tried a "let's see how deep the rabbit hole is" post! :)

And, in any normed vector space, you can split the properties of a vector into "magnitude" and "everything else", then label "everything else" as "direction"! :p~ (yes, not all vector spaces have a norm).
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Re: On Linear Algebra

Postby jtniehof » Thu Nov 08, 2007 10:37 pm UTC

Herman wrote:That sounds like the book I'm using in the LA class I have right now. (Weeeirrd...)

Linear Algebra and its Applications
Gilbert Strang
4th Ed.
ISBN 0030105676

Hmmm. Thanks, but I don't think that's it...the 4th edition came out in 2005, and the cover for the 3rd isn't familiar.

(Moral lesson: ALWAYS write down the name and author of all of your textbooks. MIT requires a list of every textbook you used when you apply for grad school...that burned me, bad.)

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Re: On Linear Algebra

Postby Token » Thu Nov 08, 2007 10:38 pm UTC

Yakk wrote::p Yes, I'm aware. But I was attempting to generate a "basic walkthrough" of what Linear Algebra was about. I already tried a "let's see how deep the rabbit hole is" post! :)

And, in any normed vector space, you can split the properties of a vector into "magnitude" and "everything else", then label "everything else" as "direction"! :p~ (yes, not all vector spaces have a norm).

Well, I kind of guessed you were aware, given that from previous forum experience you seem to know more than I do about such things, but there's simplification and there's over-simplification to the point of misleading inaccuracy. It was more a "Wait a sec, did you think that through?" than a "You could not be more wrong and I will assert my superiority by snobbishly correcting you in a condescending manner. Get your act together or be forced to get an arts degree." kind of comment.
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Re: On Linear Algebra

Postby Yakk » Thu Nov 08, 2007 11:27 pm UTC

Hey! Some of my best friends have arts degrees.

... well, no, not really.

Would you believe some friendly acquaintances?
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Re: On Linear Algebra

Postby merc » Fri Nov 09, 2007 6:15 pm UTC

jtniehof wrote:Piggy-backing:
I seem to have lost my linear algebra text, which I thought was darn good and would like to buy another copy. Does a white text with a quilt-like pattern on the cover, fairly small form factor, ring a bell for anyone? It'd be published before 1999 at the latest. (The title is completely nondescript.)

Sounds like "Linear Algebra" (or something of that nature) by Hoffman and Kunze.

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Re: On Linear Algebra

Postby wirehead » Fri Nov 09, 2007 7:29 pm UTC

You know, I keep thinking that they need to find a way to slide some amount of Linear Algebra in at around the same time as Calculus -- so, late high school or early college. I think it would be good to teach people about what ELSE is there in math besides basic geometry, algebra, and calculus and make some classes a little easier to explain mathematically. More thinking, less rote.

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Re: On Linear Algebra

Postby gmalivuk » Fri Nov 09, 2007 11:14 pm UTC

wirehead wrote:You know, I keep thinking that they need to find a way to slide some amount of Linear Algebra in at around the same time as Calculus -- so, late high school or early college.

Did you not do any linalg when you did other algebra? I distinctly remember my Algebra 2 book having matricies in it. Nothing too too complicated, but we learned to use them to solve linear equations more quickly and to invert simple ones and so on.
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Re: On Linear Algebra

Postby aguacate » Sat Nov 10, 2007 7:50 am UTC

For everyone who says you don't need any calculus to do linear algebra, have you ever heard of the wronskian? Not only did I cover it in my linear algebra class, but the students I tutor are learning it, at a separate university, in their linear algebra class.

Admittedly it's not that major of a topic in a first course in linear algebra, and it's just taking nth derivatives so it doesn't require a lot of calculus.

I thought jacobians were more a topic of multivariable calculus rather than pure linear algebra.
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Re: On Linear Algebra

Postby Buttons » Sat Nov 10, 2007 2:23 pm UTC

gmalivuk wrote:Did you not do any linalg when you did other algebra? I distinctly remember my Algebra 2 book having matricies in it. Nothing too too complicated, but we learned to use them to solve linear equations more quickly and to invert simple ones and so on.

And some classes do even more; one friend of mine learned Cramer's Rule in her 10th grade algebra class, which I find a bit ridiculous.

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Re: On Linear Algebra

Postby Owehn » Sat Nov 10, 2007 5:39 pm UTC

Don't people usually have algebra before high school? I thought typical freshman-year math was algebra II or precalculus.
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Re: On Linear Algebra

Postby Token » Sat Nov 10, 2007 5:58 pm UTC

Owehn wrote:Don't people usually have algebra before high school? I thought typical freshman-year math was algebra II or precalculus.

Elementary algebra, probably. Linear algebra, not so much.
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Re: On Linear Algebra

Postby Zohar » Sat Nov 10, 2007 6:16 pm UTC

aguacate wrote:For everyone who says you don't need any calculus to do linear algebra, have you ever heard of the wronskian? Not only did I cover it in my linear algebra class, but the students I tutor are learning it, at a separate university, in their linear algebra class.

Admittedly it's not that major of a topic in a first course in linear algebra, and it's just taking nth derivatives so it doesn't require a lot of calculus.

I thought jacobians were more a topic of multivariable calculus rather than pure linear algebra.


At our university this was part of the differential equation course (and the Jacobian for advanced calculus). But it's not an integral part of linear algebra, it's just an implementation of it.
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Re: On Linear Algebra

Postby mirashii » Sun Nov 11, 2007 3:22 pm UTC

I haven't seen this mentioned anywhere else, so I thought it would be important to mention it.

Your conceptions on Calculus and how it translates to most colleges isn't necessarily correct. AP Calculus AB should translate to all of Calculus I and about half of Calculus II at a college. Calculus BC should translate to about the later half of Calculus I and all of Calculus II at a college, maybe even getting into a tiny bit of Calculus III, but I would certainly not recommend trying to skip Calculus III after only BC.

http://www.collegeboard.com/student/tes ... topic.html

Calc BC covers up through series and some specific types of series, Taylor especially. Calc BC does some stuff with parametric and vector valued functions, but will definitely not get you nearly the exposure to these types of functions that you need in a college Calc III course, and afaik it never gets as deep into these as you may need, finding tangent lines, planes, distances from a line to plane, point to plane, etc. BC also never gets into multivariate functions and all the related topics that come along with it.

In short, don't skip Calc III.

However, you can definitely take Linear Algebra without having gone through Calc I,II,III, and IV, although I'd recommend some background in calculus so you at least know what you're doing when systems of differential equations are used in the course as a prime example and reason you learn linear algebra.

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Re: On Linear Algebra

Postby Yakk » Sun Nov 11, 2007 9:01 pm UTC

I don't understand why you think there is that much standardization in university courses.

Looking at two "calc1" courses offered at the same university, I see a huge range in the amount of content. I can't believe that there is sufficient world-wide or country-wide standardization.

If the course says it needs calc3, go and talk to the undergraduate advisors at the school and see why they need calc3 for that course.
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Re: On Linear Algebra

Postby LoopQuantumGravity » Sun Nov 11, 2007 10:00 pm UTC

Yakk wrote:I don't understand why you think there is that much standardization in university courses.

Looking at two "calc1" courses offered at the same university, I see a huge range in the amount of content. I can't believe that there is sufficient world-wide or country-wide standardization.

If the course says it needs calc3, go and talk to the undergraduate advisors at the school and see why they need calc3 for that course.


Calculus courses are pretty standard. The syllabuses for the classes may look different, since they like to emphasize different things and all, but there isn't actually that much to cover in intro calculus (despite what the 2000+ page text books would like you to believe), so things are really pretty standard.
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Re: On Linear Algebra

Postby Yakk » Mon Nov 12, 2007 3:04 am UTC

I dunno -- no mention of construction of the reals from the Peano axioms, which is what I did in calc1 if I remember right. (Plus FTC, integration from first principles, etc).

And I'm pretty certain the engineers didn't do that.
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Re: On Linear Algebra

Postby LoopQuantumGravity » Mon Nov 12, 2007 3:08 am UTC

Yakk wrote:I dunno -- no mention of construction of the reals from the Peano axioms, which is what I did in calc1 if I remember right. (Plus FTC, integration from first principles, etc).

And I'm pretty certain the engineers didn't do that.


What kind of intro calc class makes you start with that? People would kill themselves. You're probably thinking of an into analysis class (or maybe honors calculus, if your college has that kind of thing). And for the past hundred or so years, analysis =/= calculus.
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Re: On Linear Algebra

Postby Phi » Mon Nov 12, 2007 5:14 am UTC

mirashii wrote:---Information---
In short, don't skip Calc III
---Information---

I don't think I can skip Calc. III without taking the tests (although, I seem to recall my teacher talking about how a 5 on the AP gets you out of I-III. I could have misinterpreted), but that's not to say that teachers won't cover subjects taught at that level. It actually does depend on the teacher and how much you do outside of the class (It is doubtful that I will skip Calc III, but I'm just throwing that out there). For instance, my teacher, after the AP test, starts to go into Multivariable Calculus for a month (and even tests us on it). A class should not just stop because the curriculum set by people who don't know kids is over.

Also, I was sent a link from someone I know about linear algebra, and I already know around 1/3 of the course from Algebra II. Useful, that class was. [http://ocw.mit.edu/OcwWeb/web/home/home/index.htm is the greatest thing ever]

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Re: On Linear Algebra

Postby Yakk » Mon Nov 12, 2007 3:18 pm UTC

LoopQuantumGravity wrote:
Yakk wrote:I dunno -- no mention of construction of the reals from the Peano axioms, which is what I did in calc1 if I remember right. (Plus FTC, integration from first principles, etc).

And I'm pretty certain the engineers didn't do that.


What kind of intro calc class makes you start with that? People would kill themselves. You're probably thinking of an into analysis class (or maybe honors calculus, if your college has that kind of thing). And for the past hundred or so years, analysis =/= calculus.


/shrug, it was the "top" of the 5+ first-term calculus classes the university offered (107, 117, 127, 137, 147). It was a first term course called "Calculus". There where separate first-term calc courses for various majors.

They all varied and covered a variation of topics. I, naturally, have the most experience with the one I took, and the one I tutored later in my degree. But I'm relatively sure there was a difference between the "calc for science majors" and the "calc for engineering majors" and the "calc for arts majors" courses.
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