The task is in Romanian, and it translates as follows:

(1) Show that the system of vectors
$B = \{ \mathbf{v}_1 = (1, 2, 2), \mathbf{v}_2 = (1, -1, 1), \mathbf{v}_3 = (2, 1, 2) \}$
is a basis in $\mathbb{R}^3$, and determine the coordinates of the vector $\mathbf{x} = (0, 4, 1)$ in this basis.

Solution Strategy:

1. Proving the vectors form a basis: A set of vectors forms a basis for $\mathbb{R}^3$ if:

   • The vectors are linearly independent.
   • There are exactly three vectors, which matches the dimension of $\mathbb{R}^3$.

   To check linear independence, we will construct the matrix whose rows or columns are the given vectors and check if the determinant is non-zero.

2. Determining the coordinates of $\mathbf{x}$: Once we verify that the set forms a basis, we need to express the vector $\mathbf{x} = (0, 4, 1)$ as a linear combination of the basis vectors. That is, find scalars $c_1, c_2, c_3$ such that: $\mathbf{x} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3$ This leads to a system of linear equations which we can solve to find the coordinates of $\mathbf{x}$ in the given basis.

I'll proceed with the calculations.