Directions. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises is not permitted. You may not seek help from others.
Let be the -linear map from to defined by for all where M is the matrix Represent the cokernel of as a direct sum of cyclic groups.
Find one matrix in each of the similarity classes of matrices over the field (of complex numbers) that share the characteristic polynomial .
Let be the matrix in the field Find a direct sum of companion matrices that is similar to .
Let be the rational matrix Find the sequence of successively divisible invariant factors as well as the minimal and characteristic polynomials of .
Let and be -dimensional vector spaces over a field , and let be -bilinear. The bilinear form determines a linear map , with , that is defined by
Show that is an isomorphism if and only if for given bases of and of one has .
Produce such a bilinear form in the case where and when is a -dimensional vector space over .
To what extent may one generalize the two preceding items to correct statements when is a commutative ring, free -modules of rank , and a free -module of rank ?