Question

So what all can multiplication represent in mathematics? Please elaborate:

Finding volume and area from lengths

Probability of independent events happening simultaneously

What else?

Answer 1

Finding percentages.

Answer 2

Finding the instantaneous rate of change of a composite function given the instantaneous rate of changes of its precomposed functions.

Answer 3

I feel like factorials, percentages, and probability as multiplication is all fairly related in one kind of group of thinking for using multiplication, you know? But it has a very distinct meaning since we don't think of the chance of getting 2 heads in a row as being an area of 1/4.

Answer 4

But you won't find area from lengths unless the lengths are orthogonal. In probability orthogonality is independence.

Answer 5

Seems reasonable enough. So does that mean we can define dependent probabilities with some angle and find the probability with the law of cosines? Never really thought of it like that before.

Answer 6

Co-variance measures how dependent two events / random variables are.

Answer 7

It's very similar to the dot product in that it gets 0 for independent events.

Answer 8

I probably need to start looking into the applications of linear algebra on probability and come back. I never really thought of it like that, but for factorials, it's obviously dealing with orthogonality. Thanks haha, very helpful and interesting.

Answer 9

Probability doesn't use linear algebra that much as far as I can recall. It doesn't have that many systems of equations.

Answer 10

I'm basically looking to be able to derive and understand heisenberg's matrix mechanics derivation of the indeterminacy principle, so that's sort of where I'm heading eventually. I want to understand quantum mechanics haha.

Answer 11

The math itself is probably not that hard to understand. Though I'm haven't learned it.

Answer 12

Well supposedly schrodinger's equation is a representation of the same thing, and that involves using infinite dimensional space. So that way when you take the integral of a wave function times itself, you're really doing an infinite, continuous dot product to find a probability of 1. I'm not all so sure why you need an infinite space to represent a wave function though considering as we live in 3-ish? dimensions.

Answer 13

You're trying to use the vector analogy of probability for the wave equation?

Answer 14

Well no, but I guess I'm just trying to understand the math behind it and what it really means rather than being just a monkey cranking gears. I don't want to be like those kids in calculus just doing derivatives and not knowing what I'm actually doing.

So yeah, just trying to figure it out, what's the relation? I mean if you don't know that's fine, I'm just going to go read some more.

Answer 15

I don't know how far the analogy between vectors and random variables would go.

However, the vectors certainly would not be spacial vectors. Yes, the probability density functions do take spacial inputs, but the probability itself is not inherently spacial.

Answer 16

I mean if it's independent, then it can be represented as a vector. I don't really care if that vector is in euclidian space or anything like that, I understand what it means to be linearly independent just means that one doesn't affect the other, like orthogonality in a sense. I mean, sure angle between probability vectors for dependent probability is interesting to consider I guess, but not really necessary for my understanding. Yep