1. The problem asks to write the zero vector in the space of $3 \times 4$ matrices over the field $F$. 2. The zero vector in a matrix space is the matrix where every entry is zero. 3. Since the space is $M_{3 \times 4}(F)$, the zero vector is a $3$-row by $4$-column matrix with all entries equal to zero. 4. We can write it as: $$\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ 5. This matrix acts as the additive identity in $M_{3 \times 4}(F)$, meaning for any matrix $A$ in this space, $A + 0 = A$. Final answer: $$\mathbf{0} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$