binary number field, i.e. $F=\{0,1\}$

Question

The question:

Let $F$ be a binary number field, i.e. $F=\{0,1\}$. Let $A\in F^{3\times 3}$ be given by $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{bmatrix}$. Find bases for the range and the null space of $A$, respectively.

And what does $A\in F^{3\times 3}$ mean?

Answer 1

$A\in F^{3\times3}$ often mean something like "$A$ is a member of the set $F^{3\times3}$, which represents $3\times3$ matrices with entries from $F$".

In this context, "Let $A\in F^{3\times3}$..." just means "Let $A$ be a $3\times3$ matrix whose entries are in $F$..."