1.7.11 Linear transformations

condition: The linear operator \( \widehat{A} ​​\) in the space \( \mathbb{V}_{3} \) of geometric vectors is determined by the action of the mapping \( \alpha \) to the ends of the radius vectors of points in three-dimensional space. 1) Find the matrix of the linear operator \( \widehat{A} ​​\) in a suitable basis of the space \( \mathbb{V}_{3} \), and then in the canonical basis \( i, j, k \). 2) To what point in three-dimensional space does the point with coordinates \( (1,0,0) \) go under the action of the mapping \( \alpha \)? 3) Find \( A^{n} \), where \( A \) is the matrix of the operator in the basis \( i, j, k \). Mapping \( \alpha \)-projection onto the plane \( x+y+z=0 \).