How Do These Linear Algebra Concepts Align with Definitions and Vector Spaces?

[freshlikeuhh]

I am studying for an upcoming exam and am, to this end, re-reading my textbook at a slow rate to identify anything I'm not completely certain about. Linear algebra is very cumulative and proofs require a good understanding of all definitions. So I will post some questions (for which I seek brief clarification) and I would greatly appreciate any help.

1. A list of length n is an ordered collection of n objects, as defined by my textbook. The author notes that each list, by definition, has a finite length, so "that an object that looks like(x₁, x₂, x₃,...), which might be said to have infinite, length, is not a list."

He then defines something of the form Fⁿ to be the set of all lists of length n. Later in the chapter, when he defines vector spaces, he defines $$\mathbb{F}^{\infty}$$ to be a vector space. It's not that I have a hard time believing this, as addition and scalar multiplication are defined as expected, but I don't know how to reconcile this with what is said above. How can this $$\mathbb{F}^{\infty}$$ be defined as a vector space in terms of the concept of lists (which are by definition finite)?

2. So, for all fields, we require the numbers 0 and 1 to be distinct, because we need the additive and multiplicative identity to be different; that makes me wonder over what field is {0} a vector space?

I'm inclined to say over any field, as it seems you can only construct such a vector space from within a larger one; so {0} is a subspace of all vector spaces, but it cannot be strictly be a vector space (I mean, in the sense that it's not a subspace of some other vector space).

3. A polynomial with coefficients in F is said to be a function from F to F. My textbook notes that not all vector spaces consist of lists. This example, P(F), is given, for the reason that its elements consist of functions on F, not lists. I get that with P(F) being infinite dimensional, the concept of lists is inapplicable. But would it be accurate to say that the subspaces of P(F), bounded by degree m, are lists of functions (since lists are collections of objects, which may be functions or other abstract entities).

4. In the definition of a vector space, addition and scalar multiplication are treated as functions: "By addition on V we mean a function that assigns an element u+v belonging to V to each pair of elements u,v belong to V" and likewise for scalar multiplication. There's not much confusion here, except in what sense of function should I interpret this? Is this something trivial that should be overlooked, or is there any insight to be gained through thinking about it in terms of being a function?

Thanks in advance.

 

[Landau]

How can this $$\mathbb{F}^{\infty}$$ be defined as a vector space in terms of the concept of lists (which are by definition finite)?

It cannot. The elements of F^\infty simply are not lists (some would call them infinite lists). You don't need lists to define a vector space. F^\infty is an infinite-dimensional vector space.

2. So, for all fields, we require the numbers 0 and 1 to be distinct, because we need the additive and multiplicative identity to be different; that makes me wonder over what field is {0} a vector space? I'm inclined to say over any field

Yes, over every field. Recall that a vector space is just an abelian group V under addition, plus a 'scalar multiplication' F x V -> V with some properties. But for every field F, there is one and only one 'scalar multiplication' F x {0} -> {0}, namely r.0=0 for all r in F.

3. A polynomial with coefficients in F is said to be a function from F to F.

Are you sure about this definition?

4. In the definition of a vector space, addition and scalar multiplication are treated as functions: "By addition on V we mean a function that assigns an element u+v belonging to V to each pair of elements u,v belong to V" and likewise for scalar multiplication. There's not much confusion here, except in what sense of function should I interpret this? Is this something trivial that should be overlooked, or is there any insight to be gained through thinking about it in terms of being a function?

Well, how can you NOT think of it as a function? Addition is a function
$$V\times V-> V$$
given by
$$(v,w)\mapsto v+w.$$

 

[freshlikeuhh]

Thank you very much for the reply. It has certainly cleared things up for me.

Are you sure about this definition?

Oops. Seems like I got it backwards. But polynomials can be thought of as functions; so for a polynomial of finite dimension, does it make sense to think of it as a list of functions?

 

[Landau]

A polynomial itself can be thought of as a list:
$$P(X)=a_0+a_1X+a_2X^2+...+a_nX^n$$
corresponds to the list
$$(a_0,a_1,a_2,...,a_n).$$
In this way, the space of all polynomials of degree less than or equal to n is just the space of all lists of length n+1.

 

[freshlikeuhh]

Thanks.