How to Show Linearity of a Function?

[PhysicsRock]

Homework Statement

Take the Vector space of functions from a set $\{ 0,1,...,n \}$ to a field $K$, $V_n = \text{Fun}(\{ 0,1,...,n \},K)$. The notation $j_K$ refers to $\underbrace{1_K + ... + 1_K}_{j \text{-times}}$. Let $f \in V_{n+1}$. For a function $\partial : V_{n+1} \rightarrow V_n$ and $f \in V_{n+1}$, show that $\partial$ is a linear function for $\partial : i \rightarrow (i+1)_K \cdot f(i+1)$.

Relevant Equations

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I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that $f(\alpha \cdot x) = \alpha \cdot f(x)$ and $f(x_1 + x_2) = f(x_1) + f(x_2)$. However, for this specific function, I have no idea, since there is nothing provided about $f$, so if I wanted to show the multiplicative property, I couldn't just drag anything out of $f$ without loss of generality. So I could really use some help to figure this out.

Thank you in advance.

 

[pasmith]

Homework Statement:: Take the Vector space of functions from a set $\{ 0,1,...,n \}$ to a field $K$, $V_n = \text{Fun}(\{ 0,1,...,n \},K)$. The notation $j_K$ refers to $\underbrace{1_K + ... + 1_K}_{j \text{-times}}$. Let $f \in V_{n+1}$. For a function $\partial : V_{n+1} \rightarrow V_n$ and $f \in V_{n+1}$, show that $\partial$ is a linear function for $\partial : i \rightarrow (i+1)_K \cdot f(i+1)$.
Relevant Equations::

Should this not be $$\partial(f) : i \mapsto (i+1)_K \cdot f(i+1)$$

EDIT: Also, I think we need $\partial: V_{n} \to V_{n+1}$ for $\partial(f)(n)$ to be defined.

I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that $f(\alpha \cdot x) = \alpha \cdot f(x)$ and $f(x_1 + x_2) = f(x_1) + f(x_2)$. However, for this specific function, I have no idea, since there is nothing provided about $f$, so if I wanted to show the multiplicative property, I couldn't just drag anything out of $f$ without loss of generality. So I could really use some help to figure this out.

You are trying to show that $\partial : V_{n+1} \to V_n$ is linear, ie. $\partial(f_1 + f_2) = \partial (f_1) + \partial (f_2)$ for all $f_1, f_2 \in V_{n+1}$ and $\partial(\alpha f) = \alpha \partial (f)$ for every $\alpha \in K$ and every $f \in V_{n+1}$.

The operations on $V_{n+1}$ and $V_n$ are pointwise: $(f_1 + f_2)(x) = f_1(x) + f_2(x)$, $(\alpha f)(x) = \alpha f(x)$.

 

[PhysicsRock]

Should this not be $$\partial(f) : i \mapsto (i+1)_K \cdot f(i+1)$$
You are trying to show that $\partial : V_{n+1} \to V_n$ is linear, ie. $\partial(f_1 + f_2) = \partial (f_1) + \partial (f_2)$ for all $f_1, f_2 \in V_{n+1}$ and $\partial(\alpha f) = \alpha \partial (f)$ for every $\alpha \in K$ and every $f \in V_{n+1}$.

The operations on $V_{n+1}$ and $V_n$ are pointwise: $(f_1 + f_2)(x) = f_1(x) + f_2(x)$, $(\alpha f)(x) = \alpha f(x)$.

Yes, it's supposed to be $\partial(f)$. Thank you, I forgot to state that. This helps.

 

[PhysicsRock]

Should this not be $$\partial(f) : i \mapsto (i+1)_K \cdot f(i+1)$$

EDIT: Also, I think we need $\partial: V_{n} \to V_{n+1}$ for $\partial(f)(n)$ to be defined.
You are trying to show that $\partial : V_{n+1} \to V_n$ is linear, ie. $\partial(f_1 + f_2) = \partial (f_1) + \partial (f_2)$ for all $f_1, f_2 \in V_{n+1}$ and $\partial(\alpha f) = \alpha \partial (f)$ for every $\alpha \in K$ and every $f \in V_{n+1}$.

The operations on $V_{n+1}$ and $V_n$ are pointwise: $(f_1 + f_2)(x) = f_1(x) + f_2(x)$, $(\alpha f)(x) = \alpha f(x)$.

But one thing I don't understand: how do you know that $f$ obeys the requirements of a linear function?

 

[pasmith]

But one thing I don't understand: how do you know that $f$ obeys the requirements of a linear function?

$f$ doesn't have to be linear; its domain is not even a vector space!

 

[PhysicsRock]

EDIT: Also, I think we need $\partial : V_n \rightarrow V_{n+1}$ for $\partial(f)(n)$ to be defined.

I thought about that too. The assignment though says clearly what I stated in the problem description.

 

[pasmith]

The problem statement asks you to show that $\partial$ is linear; it says nothing about $f$, other than that it is an arbitrary vector in $V_n$. Perhaps the fact that you used it in your statement of the definition of a linear function is confusing you.

 

[PhysicsRock]

The problem statement asks you to show that $\partial$ is linear; it says nothing about $f$, other than that it is an arbitrary vector in $V_n$. Perhaps the fact that you used it in your statement of the definition of a linear function is confusing you.

I pretty much only translated the assignment. I felt like including everything was helpful to receive the needed aid.

 

[PeroK]

I pretty much only translated the assignment. I felt like including everything was helpful to receive the needed aid.

When you have something as abstract as this, you need to get a grip on what everything is.

I would have called $\partial$ an operator that maps functions in $V_{n+1}$ to functions in $V_n$.

For $f \in V_{n+1}$, what does $d = \partial(f)$ look like?

We know that $d \in V_n$ must be defined for $i = 0, 1 \dots n$. Working from the definition:
$$d(0) = 1_Kf(1)$$$$d(1) = 2_Kf(2)$$$$\dots d(n) = (n+1)_Kf(n+1)$$Now that we see how $\partial$ transforms functions the fun can begin!

Let $f, g \in V_{n+1}$ and $a \in K$. We need to show that:
$$\partial(f +g) = \partial(f) + \partial(g)$$$$\partial(af) = a\partial(f)$$Perhaps that's more help than I should have given you. It seems to me that the material you're studying exceeds your current capability to think in terms of abstract mathematics.