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- 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.

Thank you in advance.