A deeper understanding of the determinate

[TackyJan]

Could someone please explain what exactly is a determinate. I don't mean the definition what I mean is conceptually what does the number represent. One thing I've always disliked about how most people teach is that they don't explain the underlying meaning behind concepts. They tend to get lost in formal definitions. My learning style however requires that I have a more fundamental understanding of what something means before I apply the rules. I remember studying the determinate in college but never really understanding what it meant.

Thank you.

 

[NoodleDurh]

I was actually having a similar problem in the past. The determinate just tells us the orientation of the space we are in... better yet, it tells us the orientation of the vectors. hmm... think of the cross product of two vectors, this guy is anti-commutative, yes? This means, $a \times b = -(b \times a)$ So, this gives us an orientation... like the kind you get when you first enter a school for the first time :) Hope that helped a little more.

P.S.
Say we are on a moebius strip, how would we know when we are on the up-side or on the flip side... with that in mind catch you on the flip-side ;)

 

[kduna]

Unfortunately there is no real intuitive "this is what the determinate is" idea as far as I know. This guy gives a pretty good run through that may help you though:

http://math.stackexchange.com/questions/668/whats-an-intuitive-way-to-think-about-the-determinant

 

[FactChecker]

The magnitude of the determinant is an indication of how much the matrix increases area (2x2 matrix), volume (3x3 matrix), etc. The sign of the determinant tells if the matrix is orientation preserving (positive) or not (negative). A 3x3 matrix with a negative determinant will map a right-hand coordinate system into a left-hand coordinate system.

 

[homeomorphic]

Unfortunately there is no real intuitive "this is what the determinate is" idea as far as I know.

I disagree, and the link you provided does discuss what it is. Actually, there are two. It's either the signed volume of the parallelopiped spanned by the column vectors or it's the factor by which the corresponding linear transformation changes volumes. That is the intuitive idea, which is perhaps a little less vivid if the matrix is bigger than 3 by 3, but still applies perfectly well by analogy. I don't see why that doesn't qualify as an intuitive "this is what the determinant is" idea. Perhaps the issue is that it's not just volume, but signed volume, but that's basically a right hand rule thing. If your column vectors are ordered so as to agree with the right hand rule, you give it a positive volume, if not, you give it negative volume. To understand this more fully, especially in higher dimensions, you'd have to think a bit harder about the idea of orientation of a vector space.

Of course, for something like a complex matrix (or other more abstract matrices), the volume definition becomes more of a stretch, so it's helpful to also have the idea of multi-linear, alternating functions in general, which are nicely motivated by the volume interpretation of determinants as a special case.

 

[kduna]

Of course, for something like a complex matrix (or other more abstract matrices), the volume definition becomes more of a stretch, so it's helpful to also have the idea of multi-linear, alternating functions in general, which are nicely motivated by the volume interpretation of determinants as a special case.

This is exactly my point. Tieing yourself down in Rⁿ with geometric interpretations just seems pointless to me. As far as the multi-linear alternating function, I don't really view this as "intuitive" although for me it has always been the easiest way to think about the determinant. The reason I didn't list it, is because for most people, even if they understand what multi-linear and alternating mean, it probably won't offer any intuition.

 

[homeomorphic]

This is exactly my point. Tieing yourself down in Rn with geometric interpretations just seems pointless to me.

It's hardly pointless. First of all, R^n is quite an important special case. The volume interpretation is essential for the change of variables formula, for example. Secondly, visualization is a memory aid. To me, I can remember most things about determinants by thinking of the volume interpretation. It makes things more vivid. For example, I can immediately remember that determinant zero implies the matrix is not invertible because 0 volume implies a degenerate parallelopiped, hence the columns must not be linearly independent, so the thing must have some kernel. If I need a more general case, I can generalize the proofs easily because I have sort of integrated the algebraic idea with the geometric one, so that they aren't too far apart in my mind. Also, I do math because it amuses me, and it amuses me largely because I can visualize a lot of it. Amusement is not just for amusement's sake. It's useful to be amused. There's no sense in restricting yourself permanently to R^n, if you want to work in more generality for whatever reason, but you can add more abstractness on as an extra layer of understanding. It's not necessary to miss out on the memory-aiding and entertaining geometry.

As far as the multi-linear alternating function, I don't really view this as "intuitive" although for me it has always been the easiest way to think about the determinant. The reason I didn't list it, is because for most people, even if they understand what multi-linear and alternating mean, it probably won't offer any intuition.

There's algebraic intuition and there's geometric intuition. Multi-linearity does have a bit of both if you've thought a lot about the different interpretations and how they relate to each other.

 

[AlephZero]

Tieing yourself down in Rⁿ with geometric interpretations just seems pointless to me.

You can turn that argument around. If determinants have an intuitive interpretation as "volumes" in Rⁿ, then they are an intuitive definition of what "volume" means in a vector space where it might not be so obvious. And the concept of "volume" might be very useful for in that space.

 

[FrankFerrese]

Here is a good paper I found that answers the OP's question.
http://www.math.unt.edu/~tushar/project ideas/monthly 96 hannah geometric determinant.pdf
Frank Ferrese