1.4.23 Matrix Transformations

The linear transformation of the three -dimensional polynomial translates the vector (a_ {i}) into the vector (b_ {i} (i = 1.2,3)); a) Show that such a transformation exists and only. 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 core and image of this transformation. e) Is the transformation diagonalized? If so, then indicate the diagonal appearance 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} ]