Number of Linear boolean-functions

Question

How many linear boolean functions are there, if we have n variable?

Answer 1

If you mean that the arity is $n$, then it's $\#\hom_{\mathbb{F}_2}(\mathbb{F}_2^n,\mathbb{F}_2)=\#(\mathbb{F}_2^{n*})=\#(\mathbb{F}_2^n)=2^n$

Explanation

• $\#$ is the symbol for the cardinality of a set

• $\mathbb{F}_2$ is the 2-element field (a.k.a. $\{0,1\}$ endowed with sum and product modulo $2$)

• $\hom_{\mathbb{F}_2}(\mathbb{F}_2^n,\mathbb{F}_2)$, or $\mathbb{F}_2^{n*}$, is the set of linear functions from the $\mathbb{F}_2$-vector space $\mathbb{F}_2^n$ to $\mathbb{F}_2$. This set can be identified with the set of $n\times1$ matrixes with entries in $\mathbb{F}_2$, and therefore has exactly $2^n$ elements.