1.4.23 Matrix transformations

A linear transformation of a three-dimensional polynomial transforms a vector ( a_{i} ) into a vector ( b_{i}(i=1,2,3) ); a) Show that such a transformation exists and is unique. b) Find the transformation matrix in the basis ( a_{1}, a_{2}, a_{3} ). c) Find the transformation matrix in the standard basis (e). d) Find the kernel and image of this transformation. e) Is the transformation diagonalizable? If yes, then indicate the diagonal form and find the basis in which the transformation matrix is ​​diagonal. [ egin{array}{lll} a_{1}=(1,-1,1), & a_{2}=(1,2,0), & a_{3}=(1,1,1), \ b_{1}=(2,-2,0), & b_{2}=(7,4,10), & b_{3}=(4,2,6) . end{array} ]