For a linear system $A x = b$ where $A$ is an $m \times n$ real matrix, which of the following conditions guarantee(s) that at least one solution exists?
1. **Rouché–Capelli Theorem (Option A)**: A linear system is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}([A|b])$ (True).
2. **Column Space Definition (Option B)**: The equation $Ax = b$ expresses $b$ as a linear combination of the columns of $A$. Hence, a solution exists if and only if $b \in \operatorname{Col}(A)$ (True).
3. **Full Row Rank (Option C)**: If $\operatorname{rank}(A) = m$, the column space spans all of $\mathbb{R}^m$. Thus $b \in \mathbb{R}^m$ is always reachable, guaranteeing a solution for every vector $b$ (True).
4. **Full Column Rank (Option D)**: $\operatorname{rank}(A) = n \le m$ guarantees uniqueness of the solution if one exists, but does not guarantee existence for arbitrary $b$ (False).