3 ms·

You may be thinking of a vector as a concrete collection of values, like a vector in R^3: [x y z]. This piece is about abstract vector spaces, their properties

You may be thinking of a vector as a concrete collection of values, like a vector in R^3: [x y z]. This piece is about abstract vector spaces, their properties (vector addition, scalar multiplication, etc.) and specifically how functions meet the definition, giving you vector spaces of functions (function spaces). So the idea is that if you two functions, f and g, and a scalar b, then you can do stuff like: f + g = g + f b(f + g) = bf + bg The existence of (-f) so you have: f + (-f) = 0 Where 0 is the zero function (which also must exist for function spaces).

noduerme 3y ago
Isn't this just describing a tautology? I was reading here earlier today about the naming of the constant for light as c, and I had a question which I was too embarrassed to ask. It is this: In e=mc^2, what are the units, and if the units aren't defined and it's just a relationship, then why specify c^2? What's the point of squaring a constant, since it's just another constant? Not that I understand a damn thing about that equation. But the idea that two functions can be thought of operating additively on a vector - or space - seems... trivial. This is where I fear I am too stupid to understand the value of this.
- chongli 3y ago
  But the idea that two functions can be thought of operating additively on a vector - or space - seems... trivial. No, the functions are not operating on a vector, the functions themselves are vectors, which means you can do things like define a linear transformation L and then you have things like this: L(af + bg) = aL(f) + bL(g) You may also define a norm on a function space (such as an L^P space [1]), ||f||_p, which maps functions to non-negative real numbers and obeys the properties of norms that we expect, such as the triangle inequality: ||f + g||_p <= ||f||_p + ||g||_p [1] https://en.wikipedia.org/wiki/Lp_space https://en.wikipedia.org/wiki/Lp_space
  - noduerme 3y ago
    Thanks. I'm trying to understand this. So take a function that returns a random shuffled deck of cards with 52! possible real number vector outputs. What is the generalized insight from treating this function itself as a vector space... is there a shortcut to monte carlo-ing a million random shuffles if you can "divide" that function by one which produces a straight flush?
    - deleted 3y ago
      [deleted]
    - zem 3y ago
      > So take a function that returns a random shuffled deck of cards with 52! possible real number vector outputs. that is not a function in the (mathematical) sense that the article is talking about. a function is a mapping from a set of inputs to a set of outputs, and the same input will always map to the same output. (in programming terms it's what you would call a "pure function")
      - noduerme 3y ago
        then remapping them by some arithmetic is tautological, isn't it?
        zem 3y ago
        not quite; in the most general sense the arguments to and result of a function (the 'domain' and 'range' in mathematical jargon) need not be numbers but any set of mathematical objects. the heavy work went into proving properties of these generalised functions that were universally true, and showing that they were isomorphic to structures built up in other branches of mathematics. as a side note, one very important technique/idea in mathematics (in general, not just in the area of functional analysis) is describing something in terms of a set of properties that is both as general as possible and as minimal as possible. for instance numbers can be added, subtracted, multiplied and divided, with "obvious" real-world interpretations. mathematicians then asked themselves what properties exactly the numbers had to possess in order for those operations to be defined, and then they proceeded to find other classes of mathematical objects that also had those properties, and suddenly we were able to "add" and "multiply" things that had no obvious physical interpretation for those operations. but since their structure was mapped to the structure of the numbers, those operations could be mechanically defined over them, and you had all sorts of mathematical tools at your disposal. here a similar thing was done with functions. there had been a lot of work put into studying the operations you could do on "vector spaces", a mathematical structure that generalised the notion of a vector as a collection of numbers. then mathematicians noticed that if you took the minimal collection of properties something needed to have in order to be a vector space, functions satisfied all those properties. and voila - everything that you could prove about vector spaces (and again, it was a whole lot) was suddenly applicable to functions as well. (why some of this seems a bit tautological is that it also follows the properties of the real numbers, and even non-mathematicians have had a lot of intuition built up about how numbers behave. but it is by no means guaranteed that every mathematical construct will have these same properties.)
- boulos 3y ago
  Assuming this is in good faith, the units are: - c: speed of light in meters per second - m: mass in kilograms - energy: joules
  - noduerme 3y ago
    yes it was in total good faith. soooo... the metric system just happens to perfectly conform to meters per second squared times kilograms equaling joules? That seems... mind-blowing since it was invented before kilograms and joules would have been interchangeable.. ??.
    - evanb 3y ago
      If you express it in a different unit system all E, m, and c will take different numerical values, but the relationship will still be true. So there’s nothing special about the metric system. When we want to discuss this kind of relationship without reference to human convention we talk about a quantity’s dimension (not geometric like 3D). A Meter and a foot both have dimensions of length. c has units of length/time. 1 kilogram and 1 gram both have units of mass. And so on. https://en.wikipedia.org/wiki/Dimensional_analysis https://en.wikipedia.org/wiki/Dimensional_analysis
      - boulos 3y ago
        Yep. This was going to be the next phase: there isn't anything special about the SI system, but it is consistent. The poster didn't seem prepared for "or any other velocity units you want". For the original poster, you could equally use c in miles per hour, mass in "pounds", and get whatever that produces for energy (I'm sure there's a name for it). This is also why Imperial units produce such amusing things as Foot-Pounds (for torque). The math all works out in the end, but you get some amusing numbers along the way.
    - joshuamorton 3y ago
      Kilograms and joules have been interchangable (well, they've had a defined relationship) for a long time though, you can calculate work in a completely Newtonian context and see that the energy requires is linearly related to the mass and quadratically to velocity (like with a spring). That the direct conversion from mass to energy follows the same shape isn't really surprising, it sort of has to. That said, the joule was only explicitly defined as kgm/s^2 in 1946 (or 1935), after Einstein and nuclear physics.
- movpasd 3y ago
  It's not tautological — to pick one of the properties, you can conceive of spaces of functions where there is no -f to a function f. For example, consider the space of all positive-valued real functions. Consider otherwise functions which take colours and output letters or the alphabet. Letters of the alphabet can't be added or subtracted, so there's no vector space structure on that. On the flip side, vector spaces can be defined not just on real numbers, but complex numbers as well, or even other sets — specifically, any "field", that is, any set with addition, subtraction, multiplication, and division defined on it in a self-consistent way. There are even finite fields; vector spaces over three are relevant in cryptography. If these rules seem silly and abstract to you (shouldn't a function just be a function?!), well, that's mathematics! By elucidating the very specific conditions under which results hold, and abstracting away all the irrelevant details, you end up with results of incredible generality.

noduerme 3y ago

Isn't this just describing a tautology? I was reading here earlier today about the naming of the constant for light as c, and I had a question which I was too embarrassed to ask. It is this: In e=mc^2, what are the units, and if the units aren't defined and it's just a relationship, then why specify c^2? What's the point of squaring a constant, since it's just another constant? Not that I understand a damn thing about that equation. But the idea that two functions can be thought of operating additively on a vector - or space - seems... trivial. This is where I fear I am too stupid to understand the value of this.

chongli 3y ago
But the idea that two functions can be thought of operating additively on a vector - or space - seems... trivial. No, the functions are not operating on a vector, the functions themselves are vectors, which means you can do things like define a linear transformation L and then you have things like this: L(af + bg) = aL(f) + bL(g) You may also define a norm on a function space (such as an L^P space [1]), ||f||_p, which maps functions to non-negative real numbers and obeys the properties of norms that we expect, such as the triangle inequality: ||f + g||_p <= ||f||_p + ||g||_p [1] https://en.wikipedia.org/wiki/Lp_space https://en.wikipedia.org/wiki/Lp_space
- noduerme 3y ago
  Thanks. I'm trying to understand this. So take a function that returns a random shuffled deck of cards with 52! possible real number vector outputs. What is the generalized insight from treating this function itself as a vector space... is there a shortcut to monte carlo-ing a million random shuffles if you can "divide" that function by one which produces a straight flush?
  - deleted 3y ago
    [deleted]
  - zem 3y ago
    > So take a function that returns a random shuffled deck of cards with 52! possible real number vector outputs. that is not a function in the (mathematical) sense that the article is talking about. a function is a mapping from a set of inputs to a set of outputs, and the same input will always map to the same output. (in programming terms it's what you would call a "pure function")
    - noduerme 3y ago
      then remapping them by some arithmetic is tautological, isn't it?
      - zem 3y ago
        not quite; in the most general sense the arguments to and result of a function (the 'domain' and 'range' in mathematical jargon) need not be numbers but any set of mathematical objects. the heavy work went into proving properties of these generalised functions that were universally true, and showing that they were isomorphic to structures built up in other branches of mathematics. as a side note, one very important technique/idea in mathematics (in general, not just in the area of functional analysis) is describing something in terms of a set of properties that is both as general as possible and as minimal as possible. for instance numbers can be added, subtracted, multiplied and divided, with "obvious" real-world interpretations. mathematicians then asked themselves what properties exactly the numbers had to possess in order for those operations to be defined, and then they proceeded to find other classes of mathematical objects that also had those properties, and suddenly we were able to "add" and "multiply" things that had no obvious physical interpretation for those operations. but since their structure was mapped to the structure of the numbers, those operations could be mechanically defined over them, and you had all sorts of mathematical tools at your disposal. here a similar thing was done with functions. there had been a lot of work put into studying the operations you could do on "vector spaces", a mathematical structure that generalised the notion of a vector as a collection of numbers. then mathematicians noticed that if you took the minimal collection of properties something needed to have in order to be a vector space, functions satisfied all those properties. and voila - everything that you could prove about vector spaces (and again, it was a whole lot) was suddenly applicable to functions as well. (why some of this seems a bit tautological is that it also follows the properties of the real numbers, and even non-mathematicians have had a lot of intuition built up about how numbers behave. but it is by no means guaranteed that every mathematical construct will have these same properties.)
boulos 3y ago
Assuming this is in good faith, the units are: - c: speed of light in meters per second - m: mass in kilograms - energy: joules
- noduerme 3y ago
  yes it was in total good faith. soooo... the metric system just happens to perfectly conform to meters per second squared times kilograms equaling joules? That seems... mind-blowing since it was invented before kilograms and joules would have been interchangeable.. ??.
  - evanb 3y ago
    If you express it in a different unit system all E, m, and c will take different numerical values, but the relationship will still be true. So there’s nothing special about the metric system. When we want to discuss this kind of relationship without reference to human convention we talk about a quantity’s dimension (not geometric like 3D). A Meter and a foot both have dimensions of length. c has units of length/time. 1 kilogram and 1 gram both have units of mass. And so on. https://en.wikipedia.org/wiki/Dimensional_analysis https://en.wikipedia.org/wiki/Dimensional_analysis
    - boulos 3y ago
      Yep. This was going to be the next phase: there isn't anything special about the SI system, but it is consistent. The poster didn't seem prepared for "or any other velocity units you want". For the original poster, you could equally use c in miles per hour, mass in "pounds", and get whatever that produces for energy (I'm sure there's a name for it). This is also why Imperial units produce such amusing things as Foot-Pounds (for torque). The math all works out in the end, but you get some amusing numbers along the way.
  - joshuamorton 3y ago
    Kilograms and joules have been interchangable (well, they've had a defined relationship) for a long time though, you can calculate work in a completely Newtonian context and see that the energy requires is linearly related to the mass and quadratically to velocity (like with a spring). That the direct conversion from mass to energy follows the same shape isn't really surprising, it sort of has to. That said, the joule was only explicitly defined as kgm/s^2 in 1946 (or 1935), after Einstein and nuclear physics.
movpasd 3y ago
It's not tautological — to pick one of the properties, you can conceive of spaces of functions where there is no -f to a function f. For example, consider the space of all positive-valued real functions. Consider otherwise functions which take colours and output letters or the alphabet. Letters of the alphabet can't be added or subtracted, so there's no vector space structure on that. On the flip side, vector spaces can be defined not just on real numbers, but complex numbers as well, or even other sets — specifically, any "field", that is, any set with addition, subtraction, multiplication, and division defined on it in a self-consistent way. There are even finite fields; vector spaces over three are relevant in cryptography. If these rules seem silly and abstract to you (shouldn't a function just be a function?!), well, that's mathematics! By elucidating the very specific conditions under which results hold, and abstracting away all the irrelevant details, you end up with results of incredible generality.