What's the purpose of using prime numbers in vector spaces?

[Fja3]

In tutorials for vector subspaces, I saw that prime numbers are frequantly used in the examples. What makes them so special in this context? Why can't these be any numbers?

For instance:

Let F be a vector subspace of vector space E

u = (x, y), u' = (x',y') ∈ F₁ and α, β ∈ ℝ
αu + βu' = (αx + βx', αy + βy')
2(αx + βx') - 3(αy + βy') = α(2x - 3y) + β(2x' - 3y') = 0

Thus, αu + βu' ∈ F₁
(Meaning αu + βu' are in vector subspace F)

 

[blamocur]

In tutorials for vector subspaces, I saw that prime numbers are frequantly used in the examples. What makes them so special in this context? Why can't these be any numbers?

For instance:

Let F be a vector subspace of vector space E

u = (x, y), u' = (x',y') ∈ F₁ and α, β ∈ ℝ
αu + βu' = (αx + βx', αy + βy')
2(αx + βx') - 3(αy + βy') = α(2x - 3y) + β(2x' - 3y') = 0

Thus, αu + βu' ∈ F₁
(Meaning αu + βu' are in vector subspace F)

Where are prime numbers in your example?

 

[Dr.Peterson]

In tutorials for vector subspaces, I saw that prime numbers are frequantly used in the examples. What makes them so special in this context? Why can't these be any numbers?

For instance:

Let F be a vector subspace of vector space E

u = (x, y), u' = (x',y') ∈ F₁ and α, β ∈ ℝ
αu + βu' = (αx + βx', αy + βy')
2(αx + βx') - 3(αy + βy') = α(2x - 3y) + β(2x' - 3y') = 0

Thus, αu + βu' ∈ F₁
(Meaning αu + βu' are in vector subspace F)

All I see is that they are using distinct, small numbers, which is sensible when giving a simple example. If prime numbers are in fact common, that may be either because a significant proportion of small numbers are prime (in comparison to larger numbers), or perhaps that they are avoiding duplication of products to prevent confusion (in more complicated examples than this).

There is no particular mathematical reason in this context to prefer prime numbers; if there is any truth to your claim, it would probably be for pedagogical reasons. And if you aren't told that you can't use any numbers, then you can't make such a conclusion: This is the fallacy of over-generalization. Examples are not proof!

 

[pka]

Where are prime numbers in your example?

I think that @Fja3 means [imath]\Large (x^{\prime},y^{\prime})[/imath]

 

[Fja3]

Where are prime numbers in your example?

See the symbol which looks like a quotation mark after the x in x'. (Maybe it's supposed to be ′ ?) That means prime number in much of the notation I've encountered.

The example is from a french video, sorry about that.

 

[topsquark]

In tutorials for vector subspaces, I saw that prime numbers are frequantly used in the examples. What makes them so special in this context? Why can't these be any numbers?

For instance:

Let F be a vector subspace of vector space E

u = (x, y), u' = (x',y') ∈ F₁ and α, β ∈ ℝ
αu + βu' = (αx + βx', αy + βy')
2(αx + βx') - 3(αy + βy') = α(2x - 3y) + β(2x' - 3y') = 0

Thus, αu + βu' ∈ F₁
(Meaning αu + βu' are in vector subspace F)

It's just a notation used to indicate one system versus another. Generally we prefer to call x in one system x in another so we use x'. Depending on what you are studying you might find subscripts: [imath]x_1[/imath], for example.

-Dan

 

[Dr.Peterson]

See the symbol which looks like a quotation mark after the x in x'. (Maybe it's supposed to be ′ ?) That means prime number in much of the notation I've encountered.

The example is from a french video, sorry about that.

Media doesn't work properly; I viewed it here.

But I don't see what you are asking about there; can you tell us what time to look at?

In any case, it looks like @pka was right about your misunderstanding. The apostrophe in x', although it is read as "x prime", has nothing to do with prime numbers. It just means "another x" -- a modification of the variable name, much like a subscript.

 

[Fja3]

Media doesn't work properly; I viewed it here.

But I don't see what you are asking about there; can you tell us what time to look at?

In any case, it looks like @pka was right about your misunderstanding. The apostrophe in x', although it is read as "x prime", has nothing to do with prime numbers. It just means "another x" -- a modification of the variable name, much like a subscript.

9.52

Note that he's talking about verifying whether the expression F₁ = {(x, y ∈ ℝ² I 2x - 3y = 0} is an example of a vector subspace or not,
whether it belongs to ℝ² or not. (So since it's squared, intuitively one might be led or misled to think it has something to do with prime numbers.)

In the video after he gives a number of examples, he gives us a moment to think about whether they're vector subspaces or not, and gives us the answers. Translation:
"Let's start with F₁. The first thing to do is to verify that there's a vector nul at the end of it. See, 2 times 0 minus 3 times 0 gives 0. So far, not too hard. Presently, it's stable by linear combinations. So, let's take two elements of example F₁. u and u_prime, hence we'll call them x, y and "x_prime", "y_prime". (The word "prime" in french means "prime".) And then, the (we'll call the) numbers α and β within ℝ. The question is, is the linear combination αu + βu_prime within ℝ? Let's calculate. That's the simplest thing. So, αu + βu_prime, by definition (the) operations on R², are
αx + βx_prime, and subsequently αy + βy_prime, mmkay? And so, to see if the nul vector appears at the end, we'll check whether two times this (αx + βx_prime) minus 3 times this (αy + βy_prime) gives zero or not. So, let's see...

He then develops them, and in the next lines, he has grouped α and β so we can see what alpha and beta each yield, in order to see why it's zero. In the closing notes, thus, F₁ is a vector subspace of ℝ².

 

[pka]

whether it belongs to ℝ² or not. (So since it's squared, intuitively one might be led or misled to think it has something to do with prime numbers.)

Fja3, I seems to that you are really challenged by basic notation
For example: [imath]\mathbb{R}^2[/imath] has absolutely nothing to do with anything being squared.
Rather it is a basic, well known, well used notation for the two dimensional real space.
[imath]\mathbb{R}^2=\{(a,b): a\in\mathbb{R}~\&~b\in\mathbb{R}\}[/imath] a set equipped with operations of addition and scalar multiplication.
[imath][/imath][imath][/imath][imath][/imath]

 

[Fja3]

Fja3, I seems to that you are really challenged by basic notation
For example: [imath]\mathbb{R}^2[/imath] has absolutely nothing to do with anything being squared.
Rather it is a basic, well known, well used notation for the two dimensional real space.
[imath]\mathbb{R}^2=\{(a,b): a\in\mathbb{R}~\&~b\in\mathbb{R}\}[/imath] a set equipped with operations of addition and scalar multiplication.
[imath][/imath][imath][/imath][imath][/imath]

Ah, better just ignore that one.

This "prime" thing pops up very frequently in all of math, especially in french materials, very confusing why they chose to call it that. I knew already that it can represent the derivative of a function, courtesy of Lagrange whose notation differs from Leibniz', but not this other meaning. (I saw that there's info about this in the Wikipedia article "Prime (symbol)")

 

[Dr.Peterson]

Note that he's talking about verifying whether the expression F₁ = {(x, y) ∈ ℝ² I 2x - 3y = 0} is an example of a vector subspace or not,
whether it belongs to ℝ² or not. (So since it's squared, intuitively one might be led or misled to think it has something to do with prime numbers.)

As @pka pointed out, the notation [imath]\mathbb{R}^2[/imath] refers not to numbers being squared, but to the set [imath]\mathbb{R}\times\mathbb{R}=\{(a,b): a\in\mathbb{R}\text{ and }b\in\mathbb{R}\}[/imath], the set of ordered pairs of real numbers. (And square numbers aren't particularly related to prime numbers anyway.)

This particular subspace is the set of all points on the line [imath]2x-3y=0[/imath] (that is, ordered pairs that satisfy that equation).

In the video after he gives a number of examples, he gives us a moment to think about whether they're vector subspaces or not, and gives us the answers. Translation:
"Let's start with F₁. The first thing to do is to verify that there's a vector nul at the end of it. See, 2 times 0 minus 3 times 0 gives 0. So far, not too hard. Presently, it's stable by linear combinations. So, let's take two elements of example F₁. u and u_prime, hence we'll call them x, y and "x_prime", "y_prime". (The word "prime" in french means "prime".) And then, the (we'll call the) numbers α and β within ℝ. The question is, is the linear combination αu + βu_prime within ℝ? Let's calculate. That's the simplest thing. So, αu + βu_prime, by definition (the) operations on R², are
αx + βx_prime, and subsequently αy + βy_prime, mmkay? And so, to see if the nul vector appears at the end, we'll check whether two times this (αx + βx_prime) minus 3 times this (αy + βy_prime) gives zero or not. So, let's see...

The elements u and u', read as "u" and "u prime" are simply two elements; as I said, "prime" is just a notation to distinguish the two points. The word "prime" has many, many meanings, undoubtedly in French as in English, and even more so in mathematics. See here:

Prime -- from Wolfram MathWorld

A symbol used to distinguish one quantity x^' ("x prime") from another related x. Prime marks are most commonly used to denote 1. Transformed coordinates, 2. Conjugate points, 3. derivatives, 4. The complement F^' of a set F, 5. As an alternate notation for transpose. Of course, the term...

mathworld.wolfram.com

A symbol used to distinguish one quantity x' ("x prime") from another related x.​

They list several more specific uses of the symbol, but here it merely means, as I said, "another x". The symbol never means "prime number".

So all he is saying is that the definition of a subspace requires that when you have two elements, any linear combination of them is still an element of the subspace. He chose to call them u = (x,y) and u' = (x', y'); he could just as well have said, for example, u = (x,y) and v = (z,w). It would mean the same thing.

 

[Dr.Peterson]

Ah, better just ignore that one.

This "prime" thing pops up very frequently in all of math, especially in french materials, very confusing why they chose to call it that. I knew already that it can represent the derivative of a function, courtesy of Lagrange whose notation differs from Leibniz', but not this other meaning. (I saw that there's info about this in the Wikipedia article "Prime (symbol)")

I had intended to mention something about why the word "prime" is used for this symbol, but forgot (and it took some time to find confirmation of my understanding of it).

Here is a source I found: https://en-academic.com/dic.nsf/enwiki/379945 :

The name "prime" is something of a misnomer. Through the early part of the 20th century, the notation x′ was read as "x prime" not because it was an x followed by a "prime symbol", but because it was the first in the series that continued with x″ ("x second") and x‴ ("x third"). It was only later, in the 1950s and 1960s, that the term "prime" began to be applied to the apostrophe-like symbol itself. Although it is now more common to pronounce x″ and x‴ as "x double prime" and "x triple prime", these are still sometimes pronounced in the old manner as "x second" and "x third".​

This supports my impression, namely that "prime" means "first" (from Latin primus), and designates, for example, the first derivative, the first in a series of units (' = feet, " = inches), the first in a series of geometric transformations of a point (A, A', A", ...), and so on.

 

[Fja3]

As @pka pointed out, the notation [imath]\mathbb{R}^2[/imath] refers not to numbers being squared, but to the set [imath]\mathbb{R}\times\mathbb{R}=\{(a,b): a\in\mathbb{R}\text{ and }b\in\mathbb{R}\}[/imath], the set of ordered pairs of real numbers. (And square numbers aren't particularly related to prime numbers anyway.)

This particular subspace is the set of all points on the line [imath]2x-3y=0[/imath] (that is, ordered pairs that satisfy that equation).

So the reason the why squared numbers and the cartesian plane share this notation (ie. the superscript), is it because "the base" is bijective? (Because there relationship between each number and its pair is bijective.)

 

[JeffM]

So the reason the why squared numbers and the cartesian plane share this notation (ie. the superscript), is it because "the base" is bijective? (Because there relationship between each number and its pair is bijective.)

As has been pointed out, it is probably best just to recognize that notation is used to mean different things in different contexts.

[imath]\mathbb R^2[/imath] means the set corresponding to the real plane, each point of which can be represented by an ordered pair of real numbers. The symbol [imath]\mathbb R[/imath] does not stand for a number but rather for a set so there is no possibility of confusing it with the meaning of [imath]R^2[/imath] where R is defined as a number.

I doubt it is helpful to strain to find some subtle argument that tries to make different conventions represent a underlying identity.

 

[Dr.Peterson]

So the reason the why squared numbers and the cartesian plane share this notation (ie. the superscript), is it because "the base" is bijective? (Because the relationship between each number and its pair is bijective.)

I'm not even sure what that means. It is definitely not true that every real number corresponds to exactly one ordered pair.

I'd say it's nothing more than an analogy between two operations denoted by "[imath]\times[/imath]". [imath]\mathbb R^2=\mathbb R\times\mathbb R[/imath] "multiplies" a set by itself, and [imath]5^2=5\times5[/imath] multiplies a number by itself. So it makes sense to use the same notation.

And the reason for the notation [imath]X\times Y[/imath] for the Cartesian product of sets (and, for that matter, for the name) is at least in part that the cardinality of the product is the product of the cardinalities -- that is, the number of pairs (x,y) is the size of X times the size of Y.