Vector space - polynomials vs. functions

[musicgold]

Homework Statement

This is not a homework problem.
I am trying to understand the difference between polynomials as vectors vs. functions as vectors.

Relevant Equations

I have two sources that show that they are treated differently when represented using vectors.

As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations.

However, what I noticed in Source #2 was that, when functions are represented as vectors, the vectors elements are the output values of the function.

1. Why are polynomials and functions represented differnetly?
2. Can a polynomial be represented by a vector the same way as a function is represented?
3. How can a vector model a function that has an infinite range?

Source #1
Source #2

Thanks

 

[HallsofIvy]

1) Polynomials are, of course, a special kind of function. The vector space of all polynomials is a subspace of the vector s.typically given as "1, x, x^2, …, x^n". The vector space or all polynomials has infinite dimension but still countable while the vector space of all functions has uncountable dimension.

2) I am not sure what you mean by "represented by a vector the same way as a function is represented". I suspect you are thinking of "vectors" as a "list of numbers" but that is not generally true. It is true that, if a vector space has finite dimension, n, there exist a basis of n vectors so that every vector in that space can be written uniquely as a linear combination of those vectors. If we understand that we are using that basis, then we can give the just the coefficients of the basis vectors rather than the linear combination itself but that requires (1) that we agree on a specified basis and (2) that the vector space be finite dimensional.

3) The range of a function is not relevant. You have already said that you accept sets of polynomials as vector spaces and any polynomial has infinite range. For another example, the differential equation $$y''- 3y'+ 2y= 0$$ has general solution $$y= Ae^{x}+ Be^{2x}$$. The set of all solutions is a two dimensional vector space having $$\{e^x, e^{2x}\}$$ as basis. Each of those functions has infinite range.

 

[PeroK]

Homework Statement: This is not a homework problem.
I am trying to understand the difference between polynomials as vectors vs. functions as vectors.
Homework Equations: I have two sources that show that they are treated differently when represented using vectors.

As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations.

However, what I noticed in Source #2 was that, when functions are represented as vectors, the vectors elements are the output values of the function.

1. Why are polynomials and functions represented differnetly?
2. Can a polynomial be represented by a vector the same way as a function is represented?
3. How can a vector model a function that has an infinite range?

Source #1
Source #2

Thanks

1. Polynomials are a special case of a function. Also, in source #2, the vectors are sequences, which are also a special type of function (with domain $\mathbb{N}$).

2. A polynomial is a function. The vector space of polynomials up to a certain degree is finite dimensional. In general, function spaces are infinite dimensional.

3. Most functions have an infinite range. What you need to do is to understand how and why a set of functions meets the axioms of a vector space.

In general, I find these odd questions and would say you are looking at things the wrong way round. If you understood the material in both sources, then there would be no questions. You seem to be starting from the premise that things should be "the same" in some unspecified way.

A vector space is any set that meets the axioms. There is no obligation that any vector space be directly related to 3D spatial vectors.

 

[RPinPA]

However, what I noticed in Source #2 was that, when functions are represented as vectors, the vectors elements are the output values of the function.

OK, I see part of your confusion. Source #1 is looking at the vector space of polynomials. A "polynomial" in this space is represented by its coefficients. All the information about it is contained in the coefficients. If I tell you the coefficients of a polynomial (in increasing order) are a0, a1, a2, then you know this polynomial is a quadratic, you know how many roots it has and whether they are real or complex, and you know other things about the polynomial as well.

A polynomial is completely described by its coefficients. A finite-length vector can be called a "polynomial". Every polynomial has an equivalent finite-length vector. Every finite-length vector corresponds to a polynomial. What we are interested in, is the coefficients, not in the list of function values.

Source #2 page 1 is an entirely different kind of vector space. The elements are arbitrary functions from $\mathbb N \rightarrow \mathbb R$. The only description we have of them is all of their values for each value of $n \in \mathbb N$. And on page 2 they describe the set of functions $\mathbb R \rightarrow \mathbb R$, which again can only be represented by its set of output values.

Now here's an important point. That second set includes polynomials. And we know that the set of polynomials is a vector space, so it is a subspace of the set of functions $\mathbb R \rightarrow \mathbb R$. So some of those functions are polynomials, which could therefore be represented by the set of output values.

But polynomials also have this simpler and much more useful representation, as the finite ordered list of coefficients. As I say, that representation is a lot more useful and let's us do all kinds of analysis. It's also simpler to understand and manipulate.

The properties of a vector space and the representation of those vectors are entirely different things. It's not at all unusual to have more than one possible representation of the same vectors.

 

[musicgold]

The properties of a vector space and the representation of those vectors are entirely different things. It's not at all unusual to have more than one possible representation of the same vectors.

Oh! I get it now. I was confused about the method of representation.

Here is my understanding of the vector space: It is a general term used to indicate any type of mathematical object that can be multiplied by numbers and added together ( column matrices, polynomials, and functions) .

Now, isn't this a very general group that includes many mathematical objects. Are there any mathematical objects that are not a vector space?

 

[WWGD]

There are relatively straightforward examples of mathematical objects that are not a vector space. The points on a sphere --not closed under addition and do not contain the origin. I would say it is less likely than not that a structure will be a vector space.

 

[Stephen Tashi]

I am trying to understand the difference between polynomials as vectors vs. functions as vectors.

You are correct that there can be a distinction! We have to worry about the definition of "polynomial". Before doing that, a basic point to remember is that defining a set (e.g. the set of 3-tuples of numbers) does not define how operations (e.g. "+") are to be performed on that set. Sometimes the operations are established by mathematical tradition, so people don't mention them.

Given the set of 3-tuples of numbers, we naturally think that adding two of them should be done by (a,b,c)+(d,e,f) = (a+d,b+e,c+f). Indeed if someone were to say "the vector space of 3-tuples", we would (by tradition) understand that this type of summation is to be used in the vector space. However, in principle, you might define a different vector space on 3-tuples by defining a different sort of summation operation. Many mathematical objects are defined with language of the form " ... is a set...together with...". It's important to keep in mind that the "together with..." part is important and, execept by tradition, not defined by merely defining the set.

Another thing to keep in mind is that mathematical language is sometimes ambiguous. To say something precise about polynomials, we have to establish what a "polynomial" is. In abstract algebra (and in the mind of many students of elementary algebra) a polynomial can be considered to be a string of symbols in certain format. In abstract algebra, we consider things like "polynomials in one indeterminate", which amounts to what students of elemenary algebra call a "polynomial in the variable x". Considering a polynomial to be his sort of thing, we can define a vector space on polynomials using the same manipulations that are taught in elementary algebra. The focus is on manipulating symbols, not on considering how the symbols can be associated with a function. From this point of view, it is natural to represent a polynomal as the n-tuple of its coefficients and treat polynomials as a vector space using the customary operations on n-tuples of numbers.

In the context of a calculus course and in most applications of mathematics, a "polynomial" is understood to be a function on the real or complex numbers. Real valued functions defined on a finite set, can be considered as a finite dimensional vector by representing each function as an n-tuple of the values takes-on, and using the customary operations on those n-tuples. A real valued function on an infinite set (e.g. the whole real number line) can be considered to be an infinite dimensional vector.

So , yes, there are two different and traditional ways to consider what "polynomial" means and two different and traditional ways to define a vector space on polynomials.

 

[WWGD]

There is a problem in defining polynomials as functions in that different functions may take the same inputs into the same outputs, e.g., as a function of $\mathbb Z_2$ into itself, the two functions $x+1$ and $x^2+1$ are indistinguishable in that sense.

 

[Mark44]

There is a problem in defining polynomials as functions in that different functions may take the same inputs into the same outputs, e.g., as a function of $\mathbb Z_2$ into itself, the two functions $x+1$ and $x^2+1$ are indistinguishable in that sense.

There is no problem in defining polynomials as functions, but maybe what you wrote isn't what you actually meant.
Regarding different functions with the same inputs mapped to the same outputs, why is this a problem? $f(x) = x + 1$ and $g(x) = \frac{x^2 - 1}{x - 1}$, where $x \in \mathbb R$, are indistinguishable, except at x = 1, but what of it?

In any case, we're veering off-topic. The thread is about polynomials considered as vectors in a vector space.

 

[WWGD]

There is no problem in defining polynomials as functions, but maybe what you wrote isn't what you actually meant.
Regarding different functions with the same inputs mapped to the same outputs, why is this a problem? $f(x) = x + 1$ and $g(x) = \frac{x^2 - 1}{x - 1}$, where $x \in \mathbb R$, are indistinguishable, except at x = 1, but what of it?

In any case, we're veering off-topic. The thread is about polynomials considered as vectors in a vector space.

Well, your examples are somewhat trivial in that you are essentially multiplying by 1 except at one point. Mine is not like that. But, like you said, we can drop it.

 

[kingali]

1) Polynomials are, of course, a special kind of function. The vector space of all polynomials is a subspace of the vector s.typically given as "1, x, x^2, …, x^n". The vector space or all polynomials has infinite dimension but still countable while the vector space of all functions has uncountable dimension.

2) I am not sure what you mean by "represented by a vector the same way as a function is represented". I suspect you are thinking of "vectors" and pinoy channel as a "list of numbers" but that is not generally true. It is true that, if a vector space has finite dimension, n, there exist a basis of n vectors so that every vector in that space can be written uniquely as a linear combination of those vectors. If we understand that we are using that basis, then we can give the just the coefficients of the basis vectors rather than the linear combination itself but that requires (1) that we agree on a specified basis and (2) that the vector space be finite dimensional.

3) The range of a function is not relevant. You have already said that you accept sets of polynomials as vector spaces and any polynomial has infinite range. For another example, the differential equation $$y''- 3y'+ 2y= 0$$ has general solution $$y= Ae^{x}+ Be^{2x}$$. The set of all solutions is a two dimensional vector space having $$\{e^x, e^{2x}\}$$ as basis. All of those functions has infinite range.

'Vector space' is actually a criminally misleading term. Why aren't we all using the more neutral term 'linear space' in our undergraduate courses and note that this is the same as 'vector space' but for psychological reasons we will reserve the latter term for tuples of real or complex numbers (behind the scenes: basis-dependent realizations of linear spaces)? I spend copious amounts of time scooping these preconceived notions out of my students' brains, something that could be avoided to a large extend by a simple switch in terminology. Many authors underestimate the power of terminology!

 

[Mark44]

'Vector space' is actually a criminally misleading term.

Criminally misleading? Well, that's a stretch.

I spend copious amounts of time scooping these preconceived notions out of my students' brains, something that could be avoided to a large extend by a simple switch in terminology.

Perhaps you could expend that copious amount of time on more productive pursuits. Your "simple switch in terminology" is not so simple -- this terminology is found in countless textbooks in linear algebra.

 

[HallsofIvy]

I believe that Texas is trying to make using the phrase "vector space" rather than "linear space" a capital offense.