Noted, xash. I was out of line and I apologize for that. I’ll be extra careful from now on.
I think Mathochist’s answer might give a little insight into how mathematicians might think differently than laypeople (to whatever extent they do).
His answer did give a sort-of physical interpretation of the determinant, but he explained it by extrapolating from 1,2, and 3-d ‘volume’ to n-dimensional ‘volume’.
To me, that impulse to generalize and extrapolate is a big part of higher mathematics – good mathemeticians find interesting analogies between one form (length) and another (area, and also volume), and generalize it (n-volume). This requires a decent understand of the form to begin with, but also helps develop a better understanding of it.
I’m not sure I can say what kinds of mental properties this requires, but I am sure it’s vital to most good mathematics.
It sounds like something a lot of people go through in mathematics.
When you’re studying matrices, you can probably “visualize” the way a matrix can scale and rotate vectors. You might even be able to “let go” a little and accept that this happens in higher dimensions even if you can’t visualize it. This, you’re probably calling “understanding”.
But, when you’re presented with an abstract concept (and eigenvalues & eigenvectors aren’t very abstract, but maybe the “determinant” is) you have a little more trouble getting you head around it, and you’re worried that you don’t “understand” it.
At some point, you need to leave that desire behind and just accept definitions and applications without regard to physicality.
That applies to linear algebra, but other branches also.
FWIW, on the determinant, my basic “Matrix Analysis” book just says, “Often in mathematics it is useful to summarize a multivariate phenomenon with a single number, and the determinant is an example of this.” That’s not a definition, but sort of a motivation. It’s not supposed to be something like a scale or a shift. It’s just supposed to be a “summary” of the matrix.
If you want a geometric motivation, what Mathochist wrote is useful, but at some point you really want to abandon your desire to visualize. And like he said, what he wrote is a consequence of what we consider to be the definiton of the determinant.
Post #6 wasn’t a fulsome enough apology? :dubious:
Especially given that it seems clear wolf_meister didn’t bother to read the post . . .
pet peeve of mine . . .
In general, the unit cube in R[sup]n[/sup] with a given basis is the set of vectors with components (with respect to that basis) all between zero and 1. That is, in the standard real plane, the unit cube is the square of points (x,y) with 0<=x,y<=1.
Actually, I possibly should have mentioned this before: remember in the change-of-variables formula for multiple integrals how the jacobian shows up, and is the determinant of the matrix of partial derivatives? What this is saying is that when you think of integrating by chopping the region up into a bunch of cubes along coordinate lines in two different ways (two different sets of variables), the transformation from one coordinate system to the other is pretty close to linear if the cubes are small enough. In that case, how do we get from the volume of one cube to the volume of the transformed cube? We multiply by the determinant of the transformation.
IMHO Mathematics was a voyage of constant discovery. OK, make that IME - I had good teachers.
Well I’d say that is a useful interpretation to shut up guys like me. I did some googling and the determinant first came up in the study of solutions of linear systems of equations, before matrix algebra was developed. I’m not sure who first defined it formally, probably Cauchy (who famously replied “because it works” when asked why he has chosen something for something else in his work) but I’m sure it was not with the goal of summarizing a matrix.
Anyway I did a bit of reading on Permutation Theory and Algebraic Equations and now I am happy to take the definition on faith. 
Is geometry “physicality”?
The more I think about Mathochist’s interpretation the more I like it, because it explains so many results, like det(a*M) = a[sup]n[/sup]*det(M) or the 2x2 matrix inversion formula.
This leads to another issue, the “deducting mantra” you could call it.
It is true that there is no arguing against solid logic (and that is a good thing). But must everything be taken at face value? For example, there are many results in Real Analysis that only become obvious when one studies Complex Analysis. It’s part of its beauty. And you don’t even need to invoke geometry to see it.
So the question of “why does it work?” does have some merit. Problem is maybe “why” has no answer sometimes (which seems a really good question, should we stop asking why?) or the answer may be out of reach.
Now I feel I can answer why the determinant pops all over the place in elementary linear analysis. It’s much more fruitful to think of it as a sort of scale factor of a Linear Transformation than by it’s definition.
But I’m reminded of number theory. It seems dicey to find intuition for everything. Maybe using elliptic curves and L-functions and weird stuff like that (maybe not!). Admitedly my study of number theory is elementary and what I do know makes perfect sense.
Surely you can all agree that definitions should be obvious, because math does not progress by chance usually.
My main goal with this thread was to see where my experience differs from accomplished mathematicians but I hope it has at least produced an interesting exchange.
Depends whom you ask. GR is definitely geometry, and actually so is a lot of what underlies modern particle physics.
As a working mathematician, actually quite a lot is taken at face value, though the face is “more than skin deep”. When Misha Kapranov tells me that the coherence laws necessary to make p-brane-integrals on the fundamental n-groupoid of a configuration space are equivalent to those arising from certain inductive limits of schemes, I accept this because
a) Kapranov knows more about this than I do,
b) he has no reason to lie to me,
c) it roughly “makes sense”, and
d) I’m confident that I could take it from “makes sense” to a proof if I really wanted to spend the time and effort.
© is obviously the most ineffable of these criteria, and it’s pure intuition, built up from years and years of looking at these sorts of things. It’s like how a chess master throws out whole classes of moves as unworkable without bothering to justify each one – he “just knows”.
Oh, we keep asking why, and generally get more and deeper answers as we go. Also, sometimes we get different answers. A differential geometer says that (basic) Fourier transforms work the way they do because of the spectrum of the Laplacian on the 1-sphere, while a representation-theorist says that they work the way they do because of the structure of the algebraic dual to the 1-torus. And no, these don’t ultimately reduce to the same thing.
Where are you learning elliptic curves without grokking determinants? As Chris Rock said, “N****! Who taught you octogon?”
Definitions should be obvious a posteriori to mathematicians sufficiently expert in the field. I can define a Hopf algebra to anyone with a first course in abstract algebra, but even then they’re far from understanding why the definition is at all useful.
I know, it’s just that physicality is an ambiguous word because geometric understanding is a valuable tool in pure maths (“of course” I hear you say). For example it is believed that what allowed Gauss to develop his theory of complex numbers was his understanding of their geometric properties (his genius helped too).
Wessel’s derivation of complex numbers from “desirable geometric properties of line segments” is also very interesting by the way. It helps to demystify the imaginary unit by shifting the viewpoint to geometry.
Very interesting. It’s what I figured more or less. But THAT is knowing maths! Not just memorizing some transform tables or knowing how to solve linear systems.
I’m not learning elliptic curves, I just mixed in some jargon for effect. It was a veiled reference to FLT, kind of tongue in cheek, saying maybe such a complicated proof brings understanding in a topic with particularly simple questions but difficult answers, only most likely it doesn’t! But I was sidetracking as you see.
That’s what I meant. Now I know I’m in that category regarding determinants.
It should also be noted that there are multiple ways of thinking of complex numbers, not all of which are necessarily equivalent, though I don’t know of a proof offhand that two generalizations disagree where both are applicable.
In particular, geometers view the geometric interpretation of i as a specified rotation as an “almost-complex” structure on a general even-dimensional manifold, while algebraists regard i as a basis element (with certain nice properties) of the algebraic completion of the reals, considered as a vector space over the reals. That is, algebraists really don’t think of i as anything more than “what you need to make sure every polynomial has a root”, and certainly not necessarily as having anything to do with rotations.
A neighbor asked me the purpose of a square root. When, in daily life, does the average person need a damn square root? Beyond basic math at the grocery store, one could live a happy life never taking a damn square root, right?
These artistic neighbors cook, sew, and does home repair - including paint and wallpaper. They can eyeball how much material they need, so the need to find an area never arises. Besides, at best, this would involve squaring, not square-rooting…even if they lived in a round room! 
The point is determinants at least have some hints of a tangible application. Kernel, range, and eigen vectors (or eigen values) is just something to accept. Hell, I had to swallow a bunch of abstract functions that I have no clue where someone dreamed it all up from, for example, ever heard of the error (erf) function? Yeah, that’s a real logical thing… Or, how about logs and why they work…adding and two logs is the same as multiplying the numbers striaght forward…weird!
Speaking of which, in high school trig, I was bombarded by all these trig proofs of trig identities. I thought this might have some practical purpose in upper calculus with trig integration problems, but hell no…never saw the stuff again, and I “is” an engineer!
Engineer: One who cannot drive a train or even ring the bell, but if the train goes off the track, guess who catches hell? LoL
- Jinx
Pretty much, yeah. In daily life, nobody really needs anything most people learn in math beyond maybe a little high-school algebra.
The kernel of a linear transformation is a lot easier to understand and apply than a determinant. All the time it’s useful to know when a function is zero, and doubly so in linear algebra. Same with range – it’s great to know when a given value can’t be the outcome of a function.
Of course it’s logical. It’s absolutely necessary to solve quote a large number of integrals, especially “Gaussian” ones. You know, the ones that show up everywhere in statistics? Tell me how you’d calculate the portion of a normally distributed sample falling between two values without using it. Go on. I dare you.
Did nobody ever tell you that the logarithm is the inverse of the exponent? Multiplying exponentials is the same as adding exponents. Weird? Hardly.
What didn’t you ever see again, the proofs or the identities? The proofs I can understand because there are much simpler ones out there than the basic trig proofs, but the identities come up all the time in any decent calc 2 or 3 class. Inverse trig substitutions, elliptic changes of variables, cylindrical and spherical coordinates…
I mentioned eigenvectors and kernels as problematic because I didn’t want to limit the discution to an explanation of determinants but these concepts are indeed much easier to grasp and don’t pose any particular difficulties.
Intuition is a good word to describe what I felt I was lacking about some concepts and the discution over determinants helped there.
I could have written the OP almost word for word. I also just finished my engineering program’s linear algebra course, and I had pretty much the same issues with what motivated most of the concepts we learned. Where the OP had trouble with determinants, however, my nemeses were eigenvalues and eigenvectors. Like Pedro, I found most of the properties of the determinant to be logical and easy to remember once I understood the geometrical interpretation given above by Mathochist (we were taught that in my course). But in the case of eigenvalues, no attempt was made to justify anything. So, as a sort of semi-hijack here, would somebody mind attempting at least a short explanation of what an eigenvalue is and why that helps anybody? Last I knew the eigenvalues of an n x n matrix A were all [symbol]l[/symbol] such that det [A - [symbol]l[/symbol]I[sub]n[/sub]] = 0, but that wasn’t very illuminating to me at the time.
If you got a matrix A and V is the set of its eigenvectors, then A maps span(V) onto itself. I don’t know for sure, but I’m guessing that span(V) is the largest set with that property.
btw, Mathochist, I’m curious: what do you consider to be the real definition of the determinant?
Think of an NxN matrix as a linear deformation of an N-dimensional space, possibly including rotations (about the origin), rescalings along various directions, and shearings. Under this deformation most vectors will change both magnitude and direction. The eigenvectors are the vectors in this space which do not change direction (other than, possibly, inverting) under this transformation; they only change their sizes.
Why is this useful? Well, it’s often easier to imagine what happens to an eigenvector, or a linear combination of eigenvectors, than it is to imagine the general case. This is especially useful for matrices that actually have a complete eigenvector basis; if you can work in a coordinate system with axes parallel to the eigenvectors then you can fairly easily envision the transformation. This is even more useful when the eigenvectors form an orthogonal basis, as with symmetric matrices.
It’s also helpful if you want to imagine more complicated functions. For example, given a matrix A, how might you calculate A[sup]100[/sup] or exp(A)? Thinking of how these matrices act on eigenvectors can help to find more efficient methods of computing these values than via straightforward matrix multiplication.
No. Two counterexamples: If A has an eigenvector with eigenvalue 0, then A will map span(V) onto a proper subspace of itself. (E.g.: A = [1 0;0 0].) If A is defective (not having a full eigenvector basis), A may map a larger space than span(V) onto itself. (E.g.: A = [1 2; 0 1].)
That’s cool A. I.. I’d be interested in that discution too. Let’s see, what can I say about eigenvectors off the top of my head… if I get anything wrong, I apologize, it’s been a while.
The definition is that an eigenvector associated with the eigenvalue k for a given matrix A is transformed into a scalar multiple of itself. That is, Ax[/] = kx**. This is a nice geometrical property for a vector to have. This equation only has nontrivial solutions if det(kI - A) = 0.
Bearing this in mind, we are motivated to consider the question “Does there exist a basis for R[sup]n[/sup] consisting of eigenvectors of A”? Because that would be the “natural” basis for the linear transformation represented by the matrix A, given the above mentioned.
If (and only if) A has n linearly independent eigenvectors the answer is yes. And under this new basis A becomes a diagonal matrix. That is, in the eigenspace (the span of the eigenvectors) a vector is transformed under A by a simple scalar multiple of its coordinates instead of more complicated linear combinations.
There are other uses of eigenvectors, some theoretical some practical but I think this gives an idea of what an eigenvector is.
Sorry, that should be Ax = kx.
Is it just me or did I totally contradict myself about three times in posts #14 and #34? Anyway nevermind that… I definitely need some sleep.