Vector Space of Functions & Matrices: Notation Explained

[sponsoredwalk]

What is the set of all functions, and as a consequence the set of all m x n matrices
supposed to look like?

S = {(x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} is a regular vector space, another notation I've seen for this is:

S = {α ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} which clearly indicates α is a vector and the operations
defined on S are:

+ : S x S → S defined by + : (α,β) ↦ α + β = (x₁,x₂,...,x₊) + (y₁,y₂,...,y₊) = ...

• : S x F → F defined by • : (α,c) ↦ (cα) = c(x₁,x₂,...,x₊) = ...Now, if we look at the set of functions we see that:

+ : S x S → S defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
• : S x F → F defined by • : (f,β) ↦ (βf)(x) = βf(x).

so would my set be

S = {(f,g,...,j)|f,g,...,j∊R}

or

S = {f ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}?

I can't get the set of functions to follow the format I've used above and I don't
think either of those sets I just described make sense tbh Then with matrices, if we look at how + & • are defined for functions and notice that a
matrix is just the function
f : (i,j) ↦ A(i,j) = A_ij
it seems reasonable that the vector space of matrices is defined in the same way as the
vector space of functions.

I am really not sure, I have a feeling that for matrices the operations are
defined along the lines of:

+ : F^{m x n} x F^{m x n} → F^{m x n} defined by + : (i,j) ↦ (A + B)(i,j) = A(i,j) + B(i,j) = A_ij + B_ij

• : F^{m x n} x F → F^{m x n} defined by • : ((i,j),β) ↦ (βA)(i,j) = βA(i,j) = βA_ij

rather than

+ : S x S → S defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
• : S x F → F defined by • : (f,β) ↦ (βf)(x) = βf(x)

because that notation suggests the m x n dimension characteristic of matrices where
the standard function notation obscures it (I think). Notice the last bit of notation
(the (f,g) ↦ (f + g)(x) stuff for matrices) really doesn't make sense eitherso I don't
think it can be along these lines.

So, to sum up I'm just asking about the notation describing the vector space of functions
and as a consequence of this the notation for the vector space of all mxn matrices.

What say you?

 

[Fredrik]

Y^X is the standard notation for the set of functions from X into Y. I think the motivation for that choice is that the cartesian product Yⁿ=Y×...×Y (n factors) can be identified with the set of functions from {0,1,...,n-1} into Y. I don't know if you're familiar with the construction of the integers in set theory, but informally it can be described as

0={}
1={0}
2={0,1}
3={0,1,2}
...
n={0,1,...,n-1}
...

So the cartesian product Yⁿ=Y×...×Y can be identified with the set of functions from n into Y.

You can define a vector space structure on the set ℝ^X by

(f+g)(x)=f(x)+g(x)
(af)(x)=af(x)

The set ℝ^X isn't a vector space on its own, but the definitions above turn it into one. More precisely, it's the triple (ℝ^X,+,s) where + is addition and s is the scalar multiplication function, that should be called a vector space.

Regarding what you actually said...I don't understand the + notation in S = {(x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}. Is it supposed to mean that the "n-tuple" has infinitely many components? The other notation S = {α ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} doesn't mean what you said, because α would be equal to one of the x_i. And what do you mean by "function"? Are you talking about functions from ℝ into ℝ? If we define a vector space on that set, we can use the fact that every vector space has a basis (a maximal linearly independent set), but the basis doesn't have to be countable, so it's not true in general that we can write

$$f=\sum_{i=1}^\infty f_i e_i$$

where the e_i are basis vectors. If we can't, we also can't represent the function as the "n-tuple" (with n equal to the cardinality of the integers) (f₁,f₂,...).

 

[sponsoredwalk]

1: At 10:11 minutes in this video you'll see this S = {α ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}
notation being used. I'm just going by what was done here.

2: The ₊ notation is just a replacement for _n, I forgot I could
write using <sub>[/ sub] on this site

3: I could have called the set S I created in my OP the triple (S,+,•) but it would
have forced me to ask how I would define the set S in that structure using this
S ={(x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} notation. What I can't understand is how the set
ℝ^X inside that algebraic structure is supposed to be written.

Is ℝ^X = {(f,g,...,j)|f,g,...,j∊R}?
Maybe ℝ^X = S = {f ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}?
Perhaps ℝ^X = {f(x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}?

4: What I mean by the set of "functions" is just that, the set of functions that form
a vector space. This is the example given in nearly every book, check Hoffman/Kunze,
Friedberg, Axler for example all do in the chapters on vector spaces. Still none specify
what this set is supposed to look like, it's just a trivial thing really, I really really doubt it
is supposed to be like any of the sets I wrote above just now but I'm just writing them
to give you an idea of what I'm looking for.</sub>

 

[Fredrik]

1: At 10:11 minutes in this video you'll see this S = {α ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}
notation being used. I'm just going by what was done here.

The ∊ appears to be a typo. It was probably meant to be an = sign, and it is at 10:30 when she shows a bit more text.

It seems that what you're looking for is just the way to define the cartesian product of two sets. Informally, I think it's OK to write X×Y={(x,y)|x∊X, y∊Y}. A special case of that would be ℝ²={(x,y)|x,y∊ℝ}. However, there are two problems with this. 1. We haven't defined what an ordered pair is. 2. There's no set theoretic axiom that says that {x|x has property P} is a set. There is however an axiom that says roughly that if S is a set, then so is {x∊S|x has property P}.

These problems are dealt with in all the standard texts on set theory. I like "Classic set theory for guided independent study" by Derek Goldrei.

3: I could have called the set S I created in my OP the triple (S,+,•)

My concern was that the first time you referred to S as a "vector space" you still hadn't mentioned that you were going to define those two functions.

Is ℝ^X = {(f,g,...,j)|f,g,...,j∊R}?
Maybe ℝ^X = S = {f ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}?
Perhaps ℝ^X = {f(x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}?

The right-hand sides of these equalities mean, respectively

• ℝⁿ, for some n.
• The set of all members of the n-tuple (x₁,x₂,...,x₊) with the property that all the members of that n-tuple are real numbers. That would be either the entire n-tuple or the empty set depending on whether all its members are real numbers or not.
• The range of f, assuming that the domain of f is ℝ.

You seem to be looking for a notation similar to ℝ²={(x,y)|x,y∊ℝ} for ℝ^X. I don't think there is one.

4: What I mean by the set of "functions" is just that, the set of functions that form
a vector space.

That doesn't answer the question. What is your definition of "function"? And what sets whose members are functions do you think you can define a vector space structure on? You clearly can't do it on the "set" of all functions. The class of all functions isn't even a set.

 

[sponsoredwalk]

Wow you know what, since this weekend and finding out about algebraic structures as
structures themselves I've discovered the important distinction between the underlying
set S in (S,+,•) and (S,+,•) as a vector space in-and-of itself. Your response just
connected the dots tbh I guess this thread is a hangover from my "past" :tongue2:
I am still studying the logic of how this is all built up, how the signature of the structure
is formed and all that but I don't think any of that is relevant to us talking about (S,+,•),
I mean in a sense it is because I think the field and identity elements should be mentioned
in our structure but hopefully we can ignore these things without it affecting my question.

So let me respond to you and we'll see if I've cleared things up a bit:

That doesn't answer the question. What is your definition of "function"? And what sets whose members are functions do you think you can define a vector space structure on? You clearly can't do it on the "set" of all functions. The class of all functions isn't even a set.

"The space of functions from a set S into a field F"

&

"The space of polynomial functions over a field F".

When I read these last week I didn't understand how they fit into the overall framework
and I was trying to stupidly describe my vector space with this fancy notation.
I kind of just assumed all functions since I didn't understand this concept properly and
this shows through in my earlier responses but judging by what you wrote here the
underlying set holds great importance and the two sentences I quoted above indicate
this distinction so hopefully I'm making more sense now.

I guess the question is, in "The space of functions from a set S into a field F" if I define
(S,+,•) where:

+ : S x S → F defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
• : S x F → F defined by • : (f,β) ↦ (βf)(x) = βf(x)

1: How would I describe the set S, which is a set containing vectors (functions) in the
notation I've been using?

Also, notice in my original post I wrote + : S x S → S but now I'm writing + : S x S → F !
Oh the conceptual errors!

S = {f,g,h...|f ∊ R} ?

That doesn't make sense because f, g etc... are vectors. Perhaps S = {f(x) | x ∊ R} ?
I think this notation accounts for f as the vector and includes variables x. You see what I'm
hinting at though & I think it's grounded in more sense this time (hopefully). It's the
underlying set I'm trying to describe, devoid of any meaning given by our axioms!



2: Notice that you said X → Y is notated as Y^X well if you look at what I've just
done the operations indicate S x S → F & S x F → F so does it become F^{S x S}
or F^{S x F} or something? Is this just conceptually wrong?

3: As for "The space of polynomial functions over a field F", I mean it's just the same where
f(x) = some polynomial... I don't see the distinction but if there is one I'd like to know!

My concern was that the first time you referred to S as a "vector space" you still hadn't mentioned that you were going to define those two functions.

I defined them:

S = {(x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} is a regular vector space, another notation I've seen for this is:

S = {α ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} which clearly indicates α is a vector and the operations
defined on S are:

+ : S x S → S defined by + : (α,β) ↦ α + β = (x₁,x₂,...,x₊) + (y₁,y₂,...,y₊) = ...

• : S x F → F defined by • : (α,c) ↦ (cα) = c(x₁,x₂,...,x₊) = ...

but my language was misleading, I shouldn't have called S a vector space it's just the
alphabet I'm using, apologies. I think you see this is a perfect example of my old ways
of thinking creeping in.

The ∊ appears to be a typo. It was probably meant to be an = sign, and it is at 10:30 when she shows a bit more text.

It seems that what you're looking for is just the way to define the cartesian product of two sets. Informally, I think it's OK to write X×Y={(x,y)|x∊X, y∊Y}. A special case of that would be ℝ²={(x,y)|x,y∊ℝ}. However, there are two problems with this. 1. We haven't defined what an ordered pair is. 2. There's no set theoretic axiom that says that {x|x has property P} is a set. There is however an axiom that says roughly that if S is a set, then so is {x∊S|x has property P}.

Ah, well spotted about the "=" I thought I had seen her use that notation more than once
in the video when I watched it but must of been wrong.

Again thank you about ℝ²={(x,y)|x,y∊ℝ} I knew this was shorthand notation
but that obscured the cartesian product idea, I hate when that happens!

Better off thinking ℝx ℝ = { α = (x,y) | (x∊ℝ) ⋀ (y∊ℝ)}

This notation is justified by the axiom of specification if we know what an ordered pair
isn't it? Still I don't understand your concern about me not defining an ordered pair in
this thread, I mean I'm hardly going to quote every axiom or theorem I need to justify
what I'm doing :tongue2:

So, I think the points beside 1:, 2: & 3: above are the only questions &
assuming the rest of my language betray's no conceptual confusion to
you I think we're almost done

 

[Fredrik]

It looks like things are getting clearer. I'm in a bit of a hurry, so I will only address the specific detail that you still seem pretty confused about.

If you want to define a vector space structure on the set of functions from S into F, where S is a set and F is a field, then you do it like this:

For each f,g in F^S, define f+g by (f+g)(x)=f(x)+g(x), for all x in S.
For each f in F^S, and each a in F, define af by (af)(x)=a(f(x)) for all x in S.

You don't define addition and scalar multiplication on S. That would be a waste of time. You do it on F^S.

I don't think there's a notation for F^S that's similar to the one we use for ℝⁿ.

 

[sponsoredwalk]

The set of polynomials of degree 3 or less: P₃ = {a₀ + a₁x₁ + a₂x₂ + a₃x₃ | a₀,...,a₃ ∊ℝ}

The set of polynomials of degree n or less: P_n = {a₀ + a₁x₁ + ... + a_nx_n | n ∊ N ⋀ a₀,...,a_n ∊ℝ}

The set of all real-valued functions of one real variable: F = {f | f : ℝ→ ℝ}.

-------------

Any problems with those definitions? I got them from the Hefferon book if you need to check.

My concerns are with making these more general without messing up.

Take P_n for example, would I be committing a grevious error if I wrote:

P_n = { v = a₀ + a₁x₁ + ... + a_nx_n | n ∊ N ⋀ a₀,...,a_n ∊ F}

and removed the "real-valued" restriction? I think just setting v equal to all that is
notational clarity/convenience and uncontroversial but how is n ∊ N justified? Is it
because N is a subset of whatever field I'm working in & polynomials are only defined on
vector spaces whose field contains the natural numbers? I mean I can't write n ∊ F as it
could be any field element then, perhaps n ∊ S where S is a subset of the field that only
takes discrete values? This is pure pedantry I'm sure you've noticed but I just want to test
the waters :tongue2:
I define the vector space to be the triple (F',P_n',°) defined by it's
regular axioms. Makes sense right? P_n is just an inert set of objects
but V = (F',P_n',°) is the vector space. Is it correct to say that
V = (F',P_n',°) is the "space" of polynomials of degree less than or equal
to n?

How about the set of all real-valued functions of one real variable: F = {f | f : ℝ→ ℝ}.

I think we could write the vector space containing this set as V = (ℝ',F',°) where

+ : F x F → F defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
° : ℝ x F → F defined by ° : (f,β) ↦ (βf)(x) = βf(x)

are defined on V = (ℝ',F',°). In V the set ℝ' is just a placeholder for the more
explicit ℝ' = (ℝ,+,•,0,1) where the unprimed ℝ is the set of reals. F' (notice
the prime!) is a placeholder for the more explicity F' = (F,+,0).
I put the primes in in my set (F',P_n',°) above as well to distinguish
between P_n and P_n' = (P_n,+,0),
also F' = (F,+,•,0,1).

So what do you think? To sum up, the questions are about Hefferon's definitions, rewriting
P_n, the natural number situation & just my general definitions.

Hopefully that's a lot clearer now!

Edit: Just to be explicit:

V = (ℝ',F',°) = ((ℝ,+,•,0,1),(F,+,0),°)

V = (F',P_n',°) = V = ((F,+,•,0,1),(P_n',+,0),°)

 

[sponsoredwalk]

Assuming that is all correct I think I can now create a vector space of matrices!

"An mxn matrix over the field F is a function A from the set of pairs of integers (i,j) s.t.
1 ≤ i ≤ m, 1 ≤ j ≤ n into the field F".

Okay, using the notation for the set of functions V = {A | A : ℕ' x ℕ'' → F} where
ℕ' & ℕ'' are primed to indicate the 1 ≤ i ≤ m, 1 ≤ j ≤ n restrictions and where A : ℕ' x ℕ'' → F
is defined by A : (i,j) ↦ A(i,j) = A_ij = (A_ij)_{m x n} we can create the vector space:

M_{m x n} = (V,F,°) = ((V,+,0),(F,+,•,0,1),°)

where V is an abelian group, F is a field & ° : F x V → V is defined by
° : (c,A) ↦ (cA)(i,j) = cA(i,j) = cA_ij = c(A_ij)_{m x n}.

 

[chiro]

Assuming that is all correct I think I can now create a vector space of matrices!

"An mxn matrix over the field F is a function A from the set of pairs of integers (i,j) s.t.
1 ≤ i ≤ m, 1 ≤ j ≤ n into the field F".

Okay, using the notation for the set of functions V = {A | A : ℕ' x ℕ'' → F} where
ℕ' & ℕ'' are primed to indicate the 1 ≤ i ≤ m, 1 ≤ j ≤ n restrictions and where A : ℕ' x ℕ'' → F
is defined by A : (i,j) ↦ A(i,j) = A_ij = (A_ij)_{m x n} we can create the vector space:
https://www.physicsforums.com/editpost.php?do=editpost&p=3135267
M_{m x n} = (V,F,°) = ((V,+,0),(F,+,•,0,1),°)

where V is an abelian group, F is a field & ° : F x V → V is defined by
° : (c,A) ↦ (cA)(i,j) = cA(i,j) = cA_ij = c(A_ij)_{m x n}.

Vector spaces are for the most part fairly simple objects. If the operations (linearity and scalar multiplication) are there, then most likely everything else will follow (the other eight axioms).

The objects in question can be as simple (ie a scalar) or as complex (matrix, some complex algebraic structure), but if you can prove the linearity condition, the rest is pretty much straightforward.

 

[Fredrik]

Assuming that is all correct I think I can now create a vector space of matrices!

"An mxn matrix over the field F is a function A from the set of pairs of integers (i,j) s.t.
1 ≤ i ≤ m, 1 ≤ j ≤ n into the field F".

Okay, using the notation for the set of functions V = {A | A : ℕ' x ℕ'' → F} where
ℕ' & ℕ'' are primed to indicate the 1 ≤ i ≤ m, 1 ≤ j ≤ n restrictions and where A : ℕ' x ℕ'' → F
is defined by A : (i,j) ↦ A(i,j) = A_ij = (A_ij)_{m x n} we can create the vector space:

M_{m x n} = (V,F,°) = ((V,+,0),(F,+,•,0,1),°)

where V is an abelian group, F is a field & ° : F x V → V is defined by
° : (c,A) ↦ (cA)(i,j) = cA(i,j) = cA_ij = c(A_ij)_{m x n}.

This notation is fine. I think it's more standard to write (V,+,°) or (V,+,°,0) instead of (V,F,°), but there's nothing wrong with how you're doing it. If you say that (V,+,°,0) is a vector space over the field F, it's not necessary to include F in the tuple.

Also note that it's perfectly fine to say that the set of m×n matrices with components in F, is the set F^m×n, and that the vector space structure on that set is the standard one. Your choice to define it as the set of functions from {1,...,m}×{1,...,n} into F does however make it easier to state the definition of matrix multiplication, especially if we use the notation A_ij instead of A(i,j) when A is such a function. The easiest way to state it is (as you know)

$$(AB)_{ij}=\sum_{k=1}^n A_{ik}B_{kj}$$

 

[sponsoredwalk]

And so ends a nearly three month quest, over two and a half just to formulate the question
properly (in terms of algebraic structures which I was trying to find but thought were a part of set theory for some reason)
and the past 2 weeks on a previous (failed attempt to formulate the question kind of) thread & then this thread

Thanks man

 

[Fredrik]

The set of polynomials of degree 3 or less: P₃ = {a₀ + a₁x₁ + a₂x₂ + a₃x₃ | a₀,...,a₃ ∊ℝ}

The set of polynomials of degree n or less: P_n = {a₀ + a₁x₁ + ... + a_nx_n | n ∊ N ⋀ a₀,...,a_n ∊ℝ}

The set of all real-valued functions of one real variable: F = {f | f : ℝ→ ℝ}.

-------------

Any problems with those definitions? I got them from the Hefferon book if you need to check.

I'm going to assume that you meant the indices on x to be exponents. I'm OK with these notations, because it's clear what they all mean. I'll just add that I don't really like it when someone calls f(x) a function. f is the function and f(x) is a member of its range. He appears to be guilty of that in the definitions of those sets of polynomials, but it's also possible that he intends them to be sets of expressions of that form, not sets of functions. This thread might help you understand the difference.

My concerns are with making these more general without messing up.

Take P_n for example, would I be committing a grevious error if I wrote:

P_n = { v = a₀ + a₁x₁ + ... + a_nx_n | n ∊ N ⋀ a₀,...,a_n ∊ F}

and removed the "real-valued" restriction? I think just setting v equal to all that is
notational clarity/convenience and uncontroversial but how is n ∊ N justified? Is it
because N is a subset of whatever field I'm working in & polynomials are only defined on
vector spaces whose field contains the natural numbers?

No, it's because polynomials have non-negative integers as exponents, and because n isn't the same integer in every polynomial. You want the set to include x² as well as x-2x³. The notation should be read as "the set of all a₀ + a₁x¹ + ... + a_nxⁿ such that n is a non-negative integer and a₀,...,a_n are members of the field". If you remove the "such that n is a non-negative integer" part, then the expression would refer to the set of polynomials of degree exactly n, where n is a specific integer specified elsewhere. So it can't possibly include both x² and x-2x³.

I don't think the "v=" at the beginning makes anything clearer. I think it looks weird actually. It should probably be interpreted as "I intend to use the symbol v to represent a member of this set", so it doesn't contribute to the definition of the set.