https://file.aiquickdraw.com/mathbot/file/17265226018045l4i9yg2.jpg

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.

Să se arate că sistemul de vectori B = {v1 = (1, 2, 2), v2 = (1, -1, 1), v3 = (2, 1, 2)} este o bază în R^3 și să se determine coordonatele vectorului x = (0, 4, 1) în această bază.

Learn how to prove that the system of vectors B = {v1 = (1, 2, 2), v2 = (1, -1, 1), v3 = (2, 1, 2)} forms a basis in R^3 and how to determine the coordinates of the vector x = (0, 4, 1) in this basis. This problem covers key concepts in linear algebra, including linear independence, basis of a vector space, and coordinate systems.

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