1. **Problem statement:** Given the set $U \subset \mathbb{R}^4$ defined by the vectors $$ \mathbf{v}_1 = \begin{pmatrix}1 \\ 0 \\ 0 \\ -1\end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix}2 \\ 1 \\ 1 \\ 0\end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix}1 \\ 1 \\ 1 \\ 1\end{pmatrix}, \quad \mathbf{v}_4 = \begin{pmatrix}1 \\ 2 \\ 3 \\ 4\end{pmatrix}. $$ Find the dimension of $U$, i.e., $\dim U$. 2. **Formula and concept:** The dimension of a subspace spanned by vectors is the number of linearly independent vectors among them. 3. **Method:** To find $\dim U$, check the linear independence of $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}$ by forming a matrix with these vectors as columns and row reducing it to find the rank. 4. **Matrix formation:** $$ A = \begin{pmatrix} 1 & 2 & 1 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 1 & 3 \\ -1 & 0 & 1 & 4 \end{pmatrix} $$ 5. **Row reduce $A$: ** - Add row 4 to row 1: $$ R_1 \to R_1 + R_4 = (1-1, 2+0, 1+1, 1+4) = (0, 2, 2, 5) $$ - New matrix: $$ \begin{pmatrix} 0 & 2 & 2 & 5 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 1 & 3 \\ -1 & 0 & 1 & 4 \end{pmatrix} $$ - Swap row 1 and row 4 to get a leading 1: $$ \begin{pmatrix} -1 & 0 & 1 & 4 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 1 & 3 \\ 0 & 2 & 2 & 5 \end{pmatrix} $$ - Multiply row 1 by $-1$: $$ R_1 \to -R_1 = (1, 0, -1, -4) $$ - New matrix: $$ \begin{pmatrix} 1 & 0 & -1 & -4 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 1 & 3 \\ 0 & 2 & 2 & 5 \end{pmatrix} $$ - Subtract row 2 from row 3: $$ R_3 \to R_3 - R_2 = (0, 0, 0, 1) $$ - Subtract $2 \times$ row 2 from row 4: $$ R_4 \to R_4 - 2R_2 = (0, 0, 0, 1) $$ - Matrix now: $$ \begin{pmatrix} 1 & 0 & -1 & -4 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$ - Subtract row 3 from row 4: $$ R_4 \to R_4 - R_3 = (0, 0, 0, 0) $$ - Final matrix: $$ \begin{pmatrix} 1 & 0 & -1 & -4 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix} $$ 6. **Rank and dimension:** The number of nonzero rows is 3, so the rank of $A$ is 3. 7. **Conclusion:** The vectors span a subspace of dimension 3. **Answer:** $\boxed{3}$, which corresponds to option (c).