1. **Stating the problem:** We have a matrix $A = \{e_{ij}\}$ with column space dimension $c(A) = 3$ and two given solution vectors: $$\mathbf{x}_1 = \begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix}, \quad \mathbf{x}_2 = \begin{bmatrix}0 \\ 1 \\ 3\end{bmatrix}$$ We want to determine how many solutions the matrix equation $A\mathbf{x} = \mathbf{b}$ has and whether these two vectors are the only solutions. 2. **Understanding the problem:** The dimension of the column space $c(A) = 3$ means the matrix $A$ has full column rank if it has 3 columns. This implies the system $A\mathbf{x} = \mathbf{b}$ is consistent for any vector $\mathbf{b}$ in $\mathbb{R}^3$. 3. **Number of solutions:** - If $A$ is a square $3 \times 3$ matrix with full rank, the system has a unique solution. - If $A$ has more columns than rows, the system may have infinitely many solutions. 4. **Are these the only solutions?** - If the system is homogeneous ($\mathbf{b} = \mathbf{0}$), the solution set forms a subspace. - If the system is non-homogeneous, the solution set is an affine subspace: a particular solution plus the null space. 5. **Conclusion:** Since $c(A) = 3$, assuming $A$ is $3 \times 3$ and full rank, the system has a unique solution for each $\mathbf{b}$. The two given vectors cannot both be solutions to the same system unless $\mathbf{b}$ differs. **Final answer:** The matrix equation $A\mathbf{x} = \mathbf{b}$ has a unique solution for each $\mathbf{b}$ if $A$ is full rank with 3 columns. Therefore, these two vectors cannot both be solutions to the same system unless the right-hand side $\mathbf{b}$ changes. Hence, these are not the only solutions in general; the solution depends on $\mathbf{b}$.