Let us consider, a set of binary rectangular matrices of finite dimensions, call the set as $T$. The cardinality of the set $T$ is $2^{mn}$ where each matrix are of order m cross n.
Suppose $S$ is a map which taken a matrix from $T$ and maps to T. is is necessary that T is linear? If so, how can we establish that? If not give some justification with example. I want to study the discrete dynamics over the set $T$.
Does the set T forms a field with respect to usual $mod_2$ addition and multiplication of matrices? I guess it is!
Not every function $S:T\to T$ is linear: a simplest counterexample is a constant function $S(M):=U$ for some nonzero matrix $U$. (A linear map must map $0$ to $0$.)
If $n\ne m$, then multiplication inside $T$ is not well defined, so $T$ is neither a ring for this case.
If $n=m>1$, then $T$ has zero divisors (and is not commutative), hence it is not a field.
(Think e.g. a pair of nonzero matrices like $\pmatrix{1&0\\0&0}$ and $\pmatrix{0&0\\0&1}$...)