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 $f$?

  • $\begingroup$ 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

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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.