How Do Vector Spaces of Linear Maps Differ from Standard Vector Spaces?

[shinobi20]

Homework Statement

1. Let $F$ ($\mathbb{R}$ or $\mathbb{C}$) be a field, let $T$ be a nonempty set, and let $V$ be the set of all functions $x: T \rightarrow F$. $~V = \mathcal{F} (T, F)$ is a vector space of linear functions. In the case that $T = \{1, 2, ..., n\}$ for some positive integer $n$, do you see any similarity between $V$ and $F^n$?


2. Let $F$ be a field and let $V$ be the set of all 'infinite sequences' $x = (a_1, a_2, a_3, ...)$ of elements of $F$. It is clear that $V$ is a vector space over $F$. Do you see any similarity between $V$ and the space $\mathcal{F} (\mathbb{Z}^+, F)$ obtained by setting $T = \mathbb{Z}^+$ as in problem 1?



3. Let $F$ be a field. If $p$ is a polynomial with coefficients in $F$, regard $p$ as a function on $F$ such that,

$$p(t) = a_0 + a_1 t + a_2 t^2 + . . . + a_n t^n$$

where $a_i$'s and $t$ are in $F$. So we can define $p: F \rightarrow F$ where $p \in \mathcal{P}$. It is clear that $\mathcal{P}$ is a vector space.

A minor variation of problem 2: consider sequences indexed by the set $N = \{0, 1, 2, 3, ...\}$ of nonnegative integers, that is, sequences

$x =(a_0, a _1, a_2, ...)$ with $a_i \in F$ for all $i \in N$. This notation is better adapted for the space of polynomials $\mathcal{P}$, explain.



4. Let $V$ be the space of infinite sequences $x =(a_0, a _1, a_2, ...)$ indexed by $N = \{0, 1, 2, 3, ...\}$. Consider the set $W$ of all $x \in V$ such that, from some index onward, the $a_i$ are all $0$, i.e. $x = (0, -3, 1, 5, 0, 0, 0, ...)$. It is easy to see that $W \subset V$. Do you see any similarity between $W$ and the space $\mathcal{P}$ of polynomial functions?

Relevant Equations

Concepts about functions
Vector space axioms

Solution

1. Based on my analysis, elements of $V$ is a map from the set of numbers $\{1, 2, ..., n\}$ to some say, real number (assuming $F = \mathbb{R}$), so that an example element of $F$ is $x(1)$. An example element of the vector space $F^n$ is $(x_1, x_2, ..., x_n)$.

From this, we can see that we can form a correspondence between $x(1)$ and the first element of $(x_1, x_2, ..., x_n)$ which is $x_1$. So we can say that $x(1) = x_1$. As a result $x: T \rightarrow F$ is really a map that gives the values of the components of the elements of $F^n$. I'm not sure if this is correct.

A natural question that arose in my mind, what is the difference between a vector space $V$ of vector valued functions of a real variable $\vec{v}: \mathbb{R} \rightarrow F^n$ and the vector space $F^n$ with element $\vec{v} = (v_1, v_2, ..., v_n)$? The first one is a vector space of linear maps $\vec{v}$. The second one is just a vector space with elements $\vec{v}$.

2. I think this is exactly the same as problem 1, where here $x(1) = a_1$, $x(2) = a_2$, and so on. So that here, $x: \mathbb{Z}^+ \rightarrow F$ is the map that gives the values of each component of the element of $V$.

3. I think we can define a series $p$ as a map $p: V \times V \rightarrow F$ where $x,y \in V$, $x =(a _1, a_2, ...)$, and $y =(t, t^2, ...)$ for some $t \in F$ such that,

$$p = x \cdot y = a_1 t + a_2 t^2 + . . . $$

which is an infinite series and can be modified (see problem 4) in order to construct a polynomial. I'm not sure about this.

4. Well, same as in problem 3, although modify the quantities such that $x =(a_0, a _1, a_2, ..., a^n)$, and $y =(0, t, t^2, ..., t^n)$ and we have

$$p = x \cdot y = a_0 + a_1 t + a_2 t^2 + . . . + a_n t^n$$

but I cannot see the similarity between $W$ and $\mathcal{P}$, the only thing similar is their result, which is a polynomial.

 

[Office_Shredder]

I think 1 and 2 are fine, and the answer is that there isn't really any difference between the functions and the tuples. For (3), thinking of p as a map from VxV is wrong - p is a function of only t, not its coefficients.

What is the smallest amount of data you need to define a polynomial p?

 

[shinobi20]

I think 1 and 2 are fine, and the answer is that there isn't really any difference between the functions and the tuples. For (3), thinking of p as a map from VxV is wrong - p is a function of only t, not its coefficients.

What is the smallest amount of data you need to define a polynomial p?

Can you explain more why there is no difference and can you expound on it?

For polynomials you need the variable and the degree of the polynomial right?

 

[Office_Shredder]

Can you explain more why there is no difference and can you expound on it?

A function on 3 elements is just a list of the 3 values it takes. Whether you dress is up as a tuple or a function isn't relevant.

For polynomials you need the variable and the degree of the polynomial right?

Let p be a polynomial. It's degree is 3 and the variable is t. Is that enough?

 

[shinobi20]

A function on 3 elements is just a list of the 3 values it takes. Whether you dress is up as a tuple or a function isn't relevant.
Let p be a polynomial. It's degree is 3 and the variable is t. Is that enough?

In addition, it needs coefficients, but how do I connect the sequence concept to form a polynomial?

 

[Office_Shredder]

So, the variable actually isn't that important. Once you pick that t is your variable, it's the same variable for all polynomials, and to specify any individual polynomial you don't need to say it.

The only thing that actually matters is the coefficients of the polynomial, which you could write down in a list that looks a lot like a sequence...

 

[shinobi20]

So, the variable actually isn't that important. Once you pick that t is your variable, it's the same variable for all polynomials, and to specify any individual polynomial you don't need to say it.

The only thing that actually matters is the coefficients of the polynomial, which you could write down in a list that looks a lot like a sequence...

So the polynomial $p$ is a function of $t$ not the coefficients which is denoted by $x \in W$ where $x = (a_0, a_1, ..., a_n, 0, 0, 0, ...)$. But how do I connect $x$ to $p$ if it cannot be a function of $x$ either?

 

[Office_Shredder]

Once you give me x, the polynomial is just $a_0 + a_1 t +...$

But the fact that it's a polynomial is just window dressing. You can describe it as a sequence of the coefficients and we both know exactly what the object is.

 

[shinobi20]

Once you give me x, the polynomial is just $a_0 + a_1 t +...$

But the fact that it's a polynomial is just window dressing. You can describe it as a sequence of the coefficients and we both know exactly what the object is.

So the gist of it is that the similarity (which is the keyword) between $W$ and $\mathcal{P}$ is that the sequence element $x \in W$ is similar to an element $p \in \mathcal{P}$ in such a way that $x$ represents the coefficients of $p$ (since the only thing needed to define a polynomial are its coefficients), and that is all.

What I thought originally was that I really need to use $x$ in a certain map to produce $p$, but what really is asked here is just to display the similarity not necessarily that they are the exact same thing, although $x$ and $p$ can be made to represent the same thing.

Is my reasoning correct?

 

[Office_Shredder]

You can make an exact map. Given a sequence of coefficients, you can write down a polynomial. That's a function. Is it linear? Is it bijective?