Step 1 :Form the system of linear equations from the matrix $A$ and vector $\mathbf{u}$: \[\begin{cases} -5x_1 - 5x_2 + 0x_3 = -7 \\ 0x_1 + 3x_2 - 7x_3 = 7 \\ 6x_1 + 3x_2 + 7x_3 = 3 \end{cases}\]
Step 2 :Form the augmented matrix: \[\left[\begin{array}{ccc|c}-5 & -5 & 0 & -7 \\ 0 & 3 & -7 & 7 \\ 6 & 3 & 7 & 3\end{array}\right]\]
Step 3 :Divide the first row by -5: \[\left[\begin{array}{ccc|c}1 & 1 & 0 & 1.4 \\ 0 & 3 & -7 & 7 \\ 6 & 3 & 7 & 3\end{array}\right]\]
Step 4 :Subtract 6 times the first row from the third row: \[\left[\begin{array}{ccc|c}1 & 1 & 0 & 1.4 \\ 0 & 3 & -7 & 7 \\ 0 & -3 & 7 & -5.4\end{array}\right]\]
Step 5 :Add the second row to the third row: \[\left[\begin{array}{ccc|c}1 & 1 & 0 & 1.4 \\ 0 & 3 & -7 & 7 \\ 0 & 0 & 0 & 1.6\end{array}\right]\]
Step 6 :The last row of the augmented matrix corresponds to the equation $0=1.6$, which is a contradiction. Therefore, the system of equations has no solution.
Step 7 :\(\boxed{\text{No, the vector } \mathbf{u} \text{ is not in the null space of matrix } A.}\)