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TackyJan

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Thank you.

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- Thread starter TackyJan
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In summary, the concept of a determinate is a mathematical tool that tells us the orientation of the space we are in, or more specifically, the orientation of the vectors in that space. It can also be thought of as a measure of how much a matrix changes the volume of the space, with the sign indicating whether it is orientation preserving or not. While it may not have a clear intuitive explanation, the volume interpretation in R^n can be a helpful way to understand determinants and their properties.

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TackyJan

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Thank you.

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- #2

NoodleDurh

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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 ;)

- #3

kduna

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

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FactChecker

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homeomorphic

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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.

- #6

kduna

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homeomorphic said: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 R

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homeomorphic

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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.

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AlephZero

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kduna said:Tieing yourself down in R^{n}with geometric interpretations just seems pointless to me.

You can turn that argument around. If determinants have an intuitive

- #9

FrankFerrese

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http://www.math.unt.edu/~tushar/project ideas/monthly 96 hannah geometric determinant.pdf

Frank Ferrese

Determinism is the philosophical belief that all events, including human actions, are ultimately determined by causes external to the will. This means that everything that happens is the result of prior causes and therefore, the future is predetermined.

Determinism and free will are often seen as opposing beliefs. While determinism suggests that our actions are predetermined and we have no control over them, free will argues that we have the ability to make choices and decisions that are not influenced by external factors.

There are various types of determinism, including psychological determinism, biological determinism, and environmental determinism. Psychological determinism suggests that our actions are determined by our subconscious mind, while biological determinism argues that our genes and biology determine our behavior. Environmental determinism asserts that our actions are shaped by our surroundings and external factors.

Determinism is compatible with the principles of science as it suggests that everything that happens follows a cause and effect relationship. This aligns with the scientific method of studying and understanding the natural world through observation and experimentation.

One of the main criticisms of determinism is that it undermines the concept of free will and personal responsibility. Additionally, some argue that it neglects the role of individual agency and choice in shaping our actions. There are also philosophical and scientific debates about whether determinism is actually true or not.

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