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Vector space $S$ is the set of all functions: $f:\{-1,1\}^n\rightarrow\mathbb{R}$.

There are two things I want to do: (a) determine the dimension of $S$, (b) find a basis. I am confused by the definition since it didn't say anything about if $f$ is linear or not. How can I explicitly write the expression for each $f$?

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  • $\begingroup$ What is $\mathcal{R}$? $\endgroup$ – levap Dec 4 '16 at 0:07
  • $\begingroup$ @levap sorry for the confusion. It should be $\mathbb{R}$: natrual number set $\endgroup$ – Sleepy Dec 4 '16 at 0:13
  • $\begingroup$ $\mathbb R$ denotes usually the set of reals. You said "natural" numbers. Do you mean naturals ($\mathbb N=\{1,2,...,n,...\}$) or reals ($\mathbb R$)? $\endgroup$ – zoli Dec 4 '16 at 0:18
  • $\begingroup$ @zoli@levap Sorry... It is real number set $\endgroup$ – Sleepy Dec 4 '16 at 0:20
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The set $\{ -1, 1 \}^n$ hasn't been given a structure of a vector space and it has no such structure over $\mathbb{R}$ so it doesn't make sense to ask whether $f \colon \{-1, 1 \}^n \rightarrow \mathbb{R}$ is linear or not.

The set $S$ consists of arbitrary functions $f \colon \{-1, 1 \}^n \rightarrow \mathbb{R}$ where the vector space structure on $S$ is described by the following operations

$$ (f_1 + f_2)(x_1, \dots, x_n) = f_1(x_1, \dots, x_n) + f_2(x_1, \dots, x_n), \\ (\lambda f_1)(x_1, \dots, x_n) = \lambda f_1(x_1, \dots, x_n) $$

for $f_1,f_2 \in S$ and $\lambda \in \mathbb{R}$ where the addition and the multiplication on the right hand side is done in $\mathbb{R}$.

Since the functions in $S$ are arbitrary, to describe a function you need to specify the $2^n$ values it gives on elements of $\{ -1, 1 \}^n$. For example, for $n=1$ you need to specify $f(1)$ and $f(-1)$ and for $n = 2$, you need to specify $f(1,1), f(-1,1), f(1,-1)$ and $f(-1,1)$. There are many approaches to solve your problem and I can offer a few hints/useful observations:

  1. Try to solve the problem first for $n = 1$ and $n = 2$.
  2. To show that a vector space $S$ has dimension $k$, you can provide a linear map $T \colon S \rightarrow \mathbb{R}^k$ which is an isomorphism.
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