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Let $S= \{ 1,2,\ldots,n \}$. and $V$ be the set of all functions $f: S \to \mathbb{R}$. $V$ is a vector space defined by: $$(f+ g)(x) = f(x) + g(x) \text{ and } (cf)(x)=cf(x). $$ Find a basis for $V$ and $\dim V$.

I'm not sure how to answer this question, it seems different than how I learned solving a basis, which was through a set of vectors, anyone know how I can tackle this?

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marked as duplicate by amd, user91500, TastyRomeo, JMP, hardmath Jan 23 '17 at 14:25

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    $\begingroup$ $V$ is just another way to write $\mathbf R^n$. $\endgroup$ – Bernard Jan 22 '17 at 23:33
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Define functions $\;f_i: S\to\Bbb R\;,\;\;i=1,2,...,n\;$ , as follows

$$f_i(j)=\delta_{ij}=\begin{cases}1,&i=j\\{}\\0,&i\neq j\end{cases}$$

Show $\;\{f_i\}\;$ is linearly independent and generate all the elements of $\;V\;$ .

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