Wednesday, May 14, 2008

What is a determinant?

Determinants pros/cons
Determinants, for a teacher if I may borrow a British phrase, they are a bit of a sticky wicket. For my teaching, determinants have the qualities that I find makes a topic uninviting for students:

➢ The rules for manually calculating determinants seem arbitrary, and are nearly impossible to motivate.
➢ It is not clear what the determinant means. By which I mean it is not clear what “natural” meaning might be given to a the value of a determinant, in much the same way that slope can be imbued with the natural meaning of “rate of change”.

But determinants are also tempting:

➢ Cramer’s rule presents a powerful way to solve systems of equations
➢ Matrices are a vast generalization of number systems that have far reaching applications. Can we use determinants as a way to motivate the study of these objects?

When teaching we need motivations
The first question I ask about a topic, is what might lead someone to want to know this? Why have “slope” or a “determinant”. For motivation, slope benefits from the universality of change. A recurrent theme in mathematics education from algebra to calculus is the wide range of applications of the slope concept to measure change.
Each child must see the motivation in mathematics
History of math as a motivation
What can we use to motivate determinants? Often when I find myself struggling to motivate something I will study the historical motivations of a topic. For instance, the rules of probability are often well motivated by the story of the way Fermat and Pascal were lead to discuss a game that motivated many questions about probability.
The determinant’s history may not be as storied. It begins to late in the history of mathematics to have a wonderful story attached. Professional mathematicians, not interested lay practitioners brought it forth.
Connections/Generalizations of previous curriculum
To be a topic worthy of our student’s interest it must be made to represent something meaningful in their lives, in their experience, or in the larger life of the mind. Can we at least show that a curious soul would want to see what this is all about? We can only do this if the determinant can tell us something that our students find real. Rules that utilize determinants in part, such as Cramer’s rule, are inadequate for this job because the determinant is only part of the equation. Why not make the whole thing one giant new equation? We need to start with unadorned determinants to begin with.

In the rest of this article I will try to make the leap. I will try to motivate determinants and demystify them, and I will indicate some more uses for them. The more we understand about the determinants the more our students will understand them and find even more remarkable ways to use them.

Volume and Area
Volume and Area do provide a very concrete motivation. It is clear to a student that measuring the size of objects is a fundamental question of existence. There are two formulas for area that I would like to focus on. These are the standard the equations for the area of a parallelogram and the area of a triangle.

The inadequacies of some formulas for some area






These formulas are absolutely correct and deeply flawed. The major drawback of these formulas is that they rely on a length that is not truly “part” of the shape. The height measurement would in reality be a difficult measurement to determine. How is it possible to determine that we are really are measuring an altitude?

Determinants provide another method to find area. In this version we must assume that the vertices have been given coordinates. Then it is possible to find vectors that represent the sides. These vectors not only encapsulate the length of the side but also the direction. It is therefore possible to find the area of a parallelogram or triangle from these measurements.

For instance, take the parallelogram shown below.



In this case we can compute the vector from A to B is <> and from A to D is <>. Using these vectors we can compute the determinant and from that the area.



Because we are interested in computing with vectors we can in general imagine that one of the corners of the parallelogram is at the origin. This gives us a figure like the one below.



We can then compute the area of the parallelogram by finding the area of the rectangle and then subtracting the area of the green trapezoids and orange triangles.



This shows that the area of the parallelogram can be computed from the vectors that define the parallelogram. Because triangles are half the size of a parallelogram we can compute the area of triangles similarly.



Where and are the vectors that define two adjacent sides of our shapes.

This analysis can be extended to parallelepiped in three dimensions.

Next, we look at applying the determinant to problems involving proportions.

33 comments:

Anonymous said...

this is very well done; thanks!

the line
"Where and are the vectors that define two adjacent sides of our shapes"
appears to be missing its vectors.
anyhow, in my browser.
prob'ly the angle brackets
got swallowed by the HTML.

yours in the struggle. vlorbik.

Kate said...

Thanks for this! I "taught" determinants this year in precalc, but didn't really get how they work. Coordinate geometry to the rescue again.

Matt said...

I am glad that you have found it useful. For me it is still the start. How do we present this to students? What is the best way to tell the story of determinants?

Anonymous said...

I am so glad you put this on the internet! I teach Algebra 2, Algebra 3, and College Algebra in a career tech. I was having trouble figuring out a way to relate determinants to a real "life" aspect. The students understand Cramer's rule just fine, but they want to know why we need to use determinants. They want to know the entire "why" reason. I found that when I was in school, the only thing I used determinants for was in Calculus 3 for vector space. The students will not use vectors at this point, but it is always nice to tell the students why you have to know the concrete stuff before you can apply it to more abstract thinking.

Matt said...

Even after these posts about determinants I still find them very mysterious. I think that there is a larger question about how alternating sums somehow distill essential qualities/quantities. I am thinking of the Euler characteristic for example, or if you know about homology, the boundary operator. I found the best concrete example to build from for the determinant is as the extension of proportionality that wrote about in another post. This starts to build linear independence from proportionality.

Thanks for reading,

Matt

Anonymous said...

Thank you. I was looking around the internet looking for information on the exact nature of determinants and how they relate to cross product - given that cross product yields a vector result while determinants yield scalar results. It has become clear that the magnitude of the resultant vector from the cross product is identical to the value of the determinant. I had a feeling that this is the case, yet I'm still not satisfied as to why.

In part, I believe as you do that explaining the larger implications of things is essential to thoroughly understanding the why; something that my teachers never did in high school. Maybe to help students understand the subject matter more thoroughly, it would help to give a geometric interpretation of how determinants can help in solving systems of equations, which is one thing that it is very commonly used for. I know it's something I think would clarify a lot for me, and maybe to make a big assumption, others.

Explaining how mathematicians saw that determinates could have more applications than only geometry involving vectors would greatly help bridge the gap between just using determinants and applying them.

Matt said...

Anonymous,

I think one important question for you with the connection between the cross product and determinants is: what do you think the cross-product is? What is it's physical significance to you. The cross product can be calculated through determinants, and this clearly shows the connection between the length of the cross product and the determinant.

As far as picturing determinants via their application in solving systems of equations, I think that Al Cuoco is able to do this, and that he has some published work that does a very good job with this. He sent me some things a year ago about it. I will have to try to dig them up. This is the most popular post on the blog so I think their is an audience out there that wants a better understanding of determinants.

Matt

Anonymous said...

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Sorry for offtopic

Anonymous said...

I am a student. All this time, I was searching for a way out to understand determinants.This page frankly did the needful. Now, I can better study on this subject. Before, everthing was just mechnaical not understanding what was on.

Thanks!

Anonymous said...

We can think of determinant as a sum of the (signed) matrix permutations. Then the student wants to intuit what would ever make someone come to take these crazy permutations and then sum them? What would have ever caused someone to do this the very first time? That's what we want to know.

One could begin with a 3x3 system of linear equations and it is intuitive to ask if we can determine a condition for two or more of them to be colinear:

Say the system is represented by:
a b c
d e f
g h j

For two or more of the rows to be colinear we have (a b c) = alpha*(d e f) some alpha OR (a b c) = beta*(g h j), some beta OR (d e f) = gamma*(g h j)

By substituting for alpha, beta, and gamma one can get rid of the multipliers and wind up with:
(ae=bd & af=cd & bf=ce) OR
(ah=bg & aj=cg & bj=ch) OR
(dh=eg & dj=fg & ej=fh)

So we have three possible sets of equations, we can intuit that it might be possible to multiply each equation in a set by factors to make the sets be identical. Then we'd have a set of equations we'd know is true. Once we get that we can sum that final set to get the determinant.

Matt said...

To the last anonymous commenter:

First, I love your comment. I don't think I have seen this derivation of the formula for the determinant before. It is a little confusing for me to follow, but I think I get it (I might make a new post with the details filled in a little). I have questions about how to get from the determinant equation to all of the conditions you wrote at the end.

But it is a long way to go, and kind of a hard road. It doesn't feel like the choice are well motivated at each point, and I don't think students will see this derivation as natural.

Not that I have a natural derivation.

Hence my fascination with this subject.

Matt

Anonymous said...

I've not seen this derivation anywhere. I would like to support my argument that every step is intuitive and also not difficult by filling in the details, if you have an email adress I could send it to. It will be a bit long for a blog and I won't have time before this weekend.

Michael said...

I'm responsible for the above "natural" derivation of deteminant above, for which I still need to complete the details but have not had time. I realized that if I want to continue the conversation I should not post as anonymous anymore.

I have a deep interest in what may be considered 'natural' and I'm curious how similar my definition of the the term is to yours. I take what may be considered the student's point of view and for me for a derivation to be natural means that I can answer the question would I always do that? with a "yes" for every step in the proof. It gets a little slippery because it depends a bit on the student's "toolbox", which I would like to supplement. I aim to show that the road you mention may be a tad long, but it is not hard. Incidentally long does not equal hard :)

Michael said...

I will provide a tool the student could use to get the factors I mentioned in the earlier post. I call it the Method of Term Dependency:
A resultant object contains elements which need to match or cancel.
1) Choose an element in the result object.
2) For every primitive in the element from step 1 identify the primitives in the starting object(s) upon which it depends.
3) Write the starting object(s) using only the primitives on which the final term depends (can use 0's to fill in the other terms).
4) Identify the all ther terms generated by the isolated primitives and see how they cancel. identify the attributes that determine how they cancel.

There is more than one variation of this method. The most concrete version uses known terms, another version uses potential terms. I will show an example of version 1: prove Det(A*B)=Det(A)*Det(B)

I'm going to use 2x2 matrices instead of 3x3 to save some effort but strongly recommend going through a 3x3 example:

Let A = 1 2
3 4
Let B = a b
c d
Then A*B = 1a + 2e 1b + 2d
3a + 4e 3b + 4d
Pick a term that needs to cancel in Det(AB): 1a3b

Write the dependent primitives: 1,a,3,b
Rewrite A and B using the isolated primitives:
A' = 1 0
3 0
B' = a b
0 0
Then A'*B' = 1a 1b
3a 3b
and the student see there are two terms in the det which result: 1a3b and -1b3a and they cancel each other.

Identify the attributes that cause this: 1a and 3b are in the same sub-column of the original AB.

Following this apporach with the student should easily be able to determine the conditions under which the resultant terrms cancel each other and when the cancel terms from the other side of the equation, i.e. Det(A)*Det(B)

This version is said to use known terms because 1a3b is a known term in the result we can see by inspection.

The factors I promised above will use the 2nd verion of this method: potential terms. And when I some more time I'll post how to use the method to get them.

Michael said...

Before I get into the case of potential terms I would like to do the 3x3 example for the product rule and fill in some of the details I left out, maybe it will be more clear. I'll use alpha characters for A and numerals for B to avoid subscripts.

Let matrix A =
a b c
d e f
g h j

let B =
1 2 3
4 5 6
7 8 9

Then A*B =
1a+4b+7c 2a+5b+8c 3a+6b+9c
1d+4e+7f 2d+5e+8f 3d+6e+9f
1g+4h+7j 2g+5h+8j 3g+6h+9j

by sub-column I mean for example column 2, sub-column 3 would be 8c,8f,8j as 3x1 column vector.

To choose a product term in Det(A*B) choose some permuation of row,col with any sub-columns because these are the product terms which must cancel. For example choose 1a2d6h.

So A' =
a 0 0
d 0 0
0 h 0

and B' =
1 2 0
0 0 6
0 0 0

so that A'*B' =
1a 2a 0
1d 2d 0
0 0 6h

and taking the determinant of this yields 0 because the inside determinant of
1a 2a
1d 2d
is 0.

Now we observe that a and d were in the same sub-column (1) and that 1 2 were in the same row and we deduce that unless all the rows and sum-columns are unique we're going to get Det(A'*B') = 0 because we're going to have repeating rows and/or columns in A'*B'.

That wil leave only the products with all unique rows and sub-columns and those are conincident with Det(A)*Det(B), hence Det(A*B)=Det(A)*Det(B). Since the unique rows and columns required generalizes it is also true for nxn.

So we have stepped through the method of Term Dependency as it applies to determinants and I hope by now it's clear(?)

But before moving on to the potential terms case(next post, I promise) I want to point out that the method infact motivates the product rule, that it explains how one would ever deduce the product rule and also why they might be inclined to do so. Why? because the Det(A*B) product is a sitting duck with lots of matching terms that one would assume could be a great candidate for the method if it only it were in the student's tool box. The only part of the exercise that is not driven strictly be the method and the empirical is recognizing that the remaining terms correspond to Det(A)*Det(B).

Michael said...

To use type dependency on the earlier equations:

(ae=bd & af=cd & bf=ce) OR
(ah=bg & aj=cg & bj=ch) OR
(dh=eg & dj=fg & ej=fh)

Consider that the student needs to get them to match so that they can become identical. To do that he needs to multiply each by some term but it may not be obvious which term. He has potential final terms to match. So for the ae term he has to multiply by something in the thrid row of the original system. Say he chooses g to get aeg in the first equation of row 1. But then aeg has to correspond to some term in each of the 3 rows of the OR'd equations and they all have diagonal elements, no two in the same column. But a and g are in the same column so it has to fail for row 2 above (which is using rows 1 and 3 of the original system). In this way he deduces which potential candidates can work for each factor and that each must be a permutation. So potential term dependency is different than using known terms because he does not know the final terms to cancel, he is deducing them.

I will do one or two more posts regarding my thoughts on math instructions and will try to leave my thoughts on math instructors out of them.

Michael said...

Introduction: To get some context for my thoughts on instructions some history will help. Many years ago as a math undergraduate in California I wondered why some problems had instructions, e.g. Gaussian Elimination, but that most did not. Why did all instructors treat 'how do you solve it?' and 'what's the answer'? as the same question? A sickening feeling occurred to me that the instructors may not even have researched the instructions for a given problem and I might be wasting my time using wild guessing to get the answer when a simple didactic might exist which would make solving and understanding the given problem easier by a factor of 1000. In order to not quit school, as a sophomore I needed to persuade myself this was not possible. Using reasoning typical of a sophomore I did it as follows: I had already noticed that some of the more advanced problems depended upon earlier results that were even more basic. Therefore instructions for assigned problems might also come in handy for research and therefore the instructors might be short-changing themselves along with the students by ignoring them. And they are far too intelligent to ever let that happen. While it was easy to see they did not mind cheating the students I was betting my future on the idea that they and the textbook authors at least were enlightened. That they at least cared about the math. Thus the instructions must not exist and it was just a foolish pipe dream to think that they might.

I went along with this notion long enough to graduate and for some years beyond that. But a nagging thought kept creeping into the back of my mind. Ok, so they don't exist. But wouldn't there be a reason why not? An explanation of sorts? Some property these problems had that was different than the ones that have instructions? Certainly the instructors had no explanation why the instructions did not exist and would not even entertain the question.

Quite a long time went by and I continued to be perplexed and annoyed by this question. I knew I had to tackle it myself because no one would help. No one. I mean nobody on the planet. I decided to tackle one problem which had really bugged me as a student and simply not quit on it until I could explain rigorously, like a math proof, why it had no instructions.

After I don't know how many months of on again, off again trying I got the result but it was not what I expected. I was able to get it published in a journal, though most journal editors refused to have anything to do with it:
http://www.emis.de/journals/SWJPAM/Vol1_2004/6.pdf

In fact the instructions DO exist, it was impossible for them not to exist and my original gut fear as a sophomore was correct. Had I known that then I surely would have departed that place as fast as the registrar could process me out of there. The naive sophomore had fallen victim to a syllogism and it was perhaps the most painful object lesson of my life. Never depend on other people to act in what I believe is their own best interests.

to be continued...

Michael said...

Introduction: To get some context for my thoughts on instructions some history will help. Many years ago as a math undergraduate in California I wondered why some problems had instructions, e.g. Gaussian Elimination, but that most did not. Why did all instructors treat 'how do you solve it?' and 'what's the answer'? as the same question? A sickening feeling occurred to me that the instructors may not even have researched the instructions for a given problem and I might be wasting my time using wild guessing to get the answer when a simple didactic might exist which would make solving and understanding the given problem easier by a factor of 1000. In order to not quit school, as a sophomore I needed to persuade myself this was not possible. Using reasoning typical of a sophomore I did it as follows: I had already noticed that some of the more advanced problems depended upon earlier results that were even more basic. Therefore instructions for assigned problems might also come in handy for research and therefore the instructors might be short-changing themselves along with the students by ignoring them. And they are far too intelligent to ever let that happen. While it was easy to see they did not mind cheating the students I was betting my future on the idea that they and the textbook authors at least were enlightened. That they at least cared about the math. Thus the instructions must not exist and it was just a foolish pipe dream to think that they might.

I went along with this notion long enough to graduate and for some years beyond that. But a nagging thought kept creeping into the back of my mind. Ok, so they don't exist. But wouldn't there be a reason why not? An explanation of sorts? Some property these problems had that was different than the ones that have instructions? Certainly the instructors had no explanation why the instructions did not exist and would not even entertain the question.

Quite a long time went by and I continued to be perplexed and annoyed by this question. I knew I had to tackle it myself because no one would help. No one. I mean nobody on the planet. I decided to tackle one problem which had really bugged me as a student and simply not quit on it until I could explain rigorously, like a math proof, why it had no instructions.

After I don't know how many months of on again, off again trying I got the result but it was not what I expected. I was able to get it published in a journal, though most journal editors refused to have anything to do with it:
http://www.emis.de/journals/SWJPAM/Vol1_2004/6.pdf

In fact the instructions DO exist, it was impossible for them not to exist and my original gut fear as a sophomore was correct. Had I known that then I surely would have departed that place as fast as the registrar could process me out of there. The naive sophomore had fallen victim to a syllogism and it was perhaps the most painful object lesson of my life. Never depend on other people to act in what I believe is their own best interests.

to be continued...

Michael said...

To anyone who may be following these posts I apologize for the length of time between them, I've
just been busy. I also apologize for the length of some of these.

With the small discovery I'd made came the rage inducing realization that I'd been gulled by indifferent instructors and rotten textbooks (more on rage later).

But along with rage came the sudden awareness that any known solution was a candidate for the
same experience. As a sophomore I'd labored through the referenced problem 39, getting a solution by substituting variables without having a clue why the last substition worked (i.e. all the terms cancelled) and all the previous tries did not.

To get the general solution I found myself working backwards from the solution to see if I could figure out exactly that-why the last substitution worked, what was different about it.

In the intervening years between graduation and 2004 I'd returned to school, eventually getting a masters in Software Engineering. I could not get the math out of my blood though and found that my mind kept returning to these past demons.

Upon analysis what I'd done was to reverse-engineer the solution to determine the context which
exposed the attribute(s) which made each step of the solution possible. Thus the general
solution (i.e. the instructions) was bound at the hip to the solution. So if the solution was nothing terribly exotic or even noteworthy it stood to reason that the general solution would not be either and thus the findings would never advance anyone's career. Perhaps this was why the math instructors never pursued it. Or maybe I'm giving them way too much credit.

I found the relationship between the solution and its parent intriguing and my engineering
backgrouind surely saved me from the inevitable dilemma I'd have had as a math student. You see instructions, at least the only ones I thought were worth a damn, were always programmable, from
what little I knew about programming. This algorithm wasn't programmable, it was too general,
although the application to problem 39 certainly can be coded.

But the engineer knew an abstract
class when he saw one and this was an abstract class. It does not get instantiated and serves as a template for classes that do. It also serves as a clean cutoff for programmers like myself who do not attempt self-writing code. I'd have never gotten past this point as a math student. No way. I'd have never made the distniction, simply not smart enough.

In fact Term Dependency is also an abstract class which derives from the parent class I described (reverse engineering the solution), but it is a partitioning of the approach to look at specific terms and the attribute in quesiton is just the dependent term. I know I've got it when it seems intuitive. And until it's intuitive I've got nothing, which is why I never quit on 39 when I just had the solution.

In all of this a monumental irony has not been lost on me. The results I'd found are counter-examples to the prevailing wisdom which says they cannot exist, let alone be trivial.

They cannot exist because their absence from the textbook obviously brings into question the
purpose of the course, after all it's called a course of instruction, right?. So in that sense they are interesting, at least a little. The irony is that it would actually be far more
interesting to find a solution that could be proved to have no general solution and I put the
challenge out there for anyone to accept. Pick any solution you want and prove it has no instructions.

Thus far no takers.

Is there anything more ironic than having something be problematic to prove even once, and yet is
assumed without question to be true by EVERYONE more than 90% of the time? I do mean everyone,
every last soul on the planet (oh, except for myself of course!).

next, why the rage ...

Michael said...

The math curriculum is advertised in the catalog as a course of instruction. The clear implication is that the instructions, to the extent that they exist in nature, will be provided. The presence and antiquity of the didactics in the textbook suggests that they've all long ago been discovered. The prof's advice to "play with it" to get the answer also implies that a didactic does not exist. All these things provide a fundamentally wrong view of mathematics. It does not matter to most students because they care only about the GPA or class standing. But to the truth seeker it matters greatly.

Problem 39 has unique instructions because the question "why did the last substitution work the others did not" has only one context for the answer. Thus the absence of it from the textbook or the lecture notes proves the instructors did not even look for the instructions before assigning the problem and the advice to "play with it" was disingenuous. If the truth seeker at least knows they didn't look then he'll search for and find it himself. But if the instructor misleads him into thinking it does not exist then his time is being wasted and that's the source of the rage, the total lack of intellectual honesty.

if the truth seeker knows the instructions have been ignored then staying in the class is problematic. He knows that if he wants to know the truth about a problem then he will have to derive it himself which presents two problems. If he is deriving the instructions that makes him the instructor, why take the class? And if he must stop to derive the instructions then how to keep up with the class?

Khurram said...

Amazing work. Exactly what I was looking for.

Anonymous said...

Khurram, anything specific?

Anonymous said...

Thank you for the post!! Super! Helped a lot :) I've always so far been struggling with the intuition of the determinant. Even in college professors only showed the formulas, but didn't explain what, why, when, etc x(

Unknown said...

Everything in the first half of the course can be motivated by the sum of signed products of permutations of the independent variables. Another example is the matrix of cofactors, which the textbooks define for proving matrix inverses but do not motivate, so the students are left wondering where it came from. It can be trivally derived by starting with the off diagonal zeros of the identity matrix.

sarah said...

Thanks for this...it's a start. I'm having trouble wrapping my brain around some of it. I studied pure math as an undergrad and now teach high school algebra 2. While I can follow (with some effort) the derivations in the post, I'm having a hard time translating this into something a 14 or 15 year old will connect to. Still working on it... Thanks for the help!

Matthew Bardoe said...

@Sarah... I agree that it is definitely difficult to use this to generate to something that 14 or 15 will connect to. It was an attempt on my part, and not a complete success. It is good at least to have these thoughts here for the teachers, who might find a way to put them together. It is possible that the proportions approach is more accessible...

Anonymous said...

Sarah, For the linear algebra stuff I guess the most importing thing for beginners is to understand the context of independent variables. I guess for 14 or 15 year olds the starting point in linear algebra is to get them to feel comfortable with the idea that if you assume independent variables and A+B=C+D then (A=C & B=D) || (A=D & B=C). In other words it's rules for substitution, because in linear algebra what you see is pretty much what you get, as opposed to something like splitting fields. So if they have some idea why they are doing the substitions then it is far better than how I tried to learn just by hacking away until something magically worked and the terms cancel. The substitutions and the resultant equations need to become intuitive. The process needs to feel intuitive. Perhaps the worst thing teachers do to their students (and there is just so very much to choose from) is that they separate them from their intuition.

sarah said...

Agreed...thanks. Here's a site I found with some matrix/determinant history if you're interested.

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html

Anonymous said...

Perhaps more relevant to your needs, Sarah, you could actually use the above problem 39 as something your students can connect to. The problem is really just a simple extension of high school algebra with maybe a couple of the rules changed which make using the independent varibles context handy. The symbolic matrices are really just variables which perhaps don't commute like real numbers but still associate and distribute but don't divide. You're then just adding the context of independent variables to the problem as a tool to be used because it is useful for manipulating matrices as independent variables as opposed to just manipulating real numbers. You wouldn't necessarily have to fully define nxn matrices for your students, just define them as objects with slightly different rules.

Anonymous said...

I have looked at the history of determinants and linear algebra, in trying to understand how they initially "figured out" all of this stuff, maybe there is a record which makes sense in the way I've tried to make sense of it. There is not. I think the die was cast a very long time ago. It was always incumbent upon the instructors to derive correct proofs, but it was never required for them to motivate the proofs or solutions. So invariably they end up recalling a solution they used the last time they saw the problem or a similar one. And that somehow passes for instruction.

Matthew Bardoe said...

I believe from what I have read, that the idea of this function determinant, may really have put together with its relationship with the matrix by cayley or sylvester. I am not sure. I would love to get a reference to the history of matrices, or some cayley's or sylvester's original papers. What I have tried to do here is give a feeling to the many ideas of independence. One idea is the linear algebra definition that is definitely difficult for teenager to understand. A simpler definition can be found using geometry, namely that the volume/area defined by a set of direction is non-zero, and therefore since these concepts are talking about the same idea then their associated formulas are related. Same with proportionality.

Anonymous said...

Hi! Here's a great article I found on net, that finds the formula for determinant assuming that we wish to measure volume. It gives a nice explanation how it was possibly invented. http://www.askamathematician.com/2013/05/q-why-are-determinants-defined-the-weird-way-they-are/

Unknown said...

The problem with the way determinants are taught is that the instructors do not identify and analyze the relevant assumption. The assumption is alluded to in the post on this site 12-16-09. The instructors don't know about this origin because they do not think about determinant as a statement arising from an assumption.