Linear algebra
1. Let be a field, char . For , show that there is a basis of (the ring of matrices over ) consisting only of non-diagonalizable matrices.
2. Let be a field, and a multiplicative function, i.e. for all .
i) Show that the following are equivalent:
a)
b)
c) vanishes on a proper nonempty subset of .
ii) If the conditions in (i) are satisfied, show that the vanishing set of is precisely the set of singular matrices, i.e. matrices with determinant 0 . Deduce that the set of maps satisfying (i) is in bijection with group homomorphisms from to .
3. i) Let be an endomorphism of a finite-dimensional -vector space. Suppose satisfy . Show that .
ii) Show that one can always satisfy the hypotheses of (i) by taking for some , .
4. Let be a vector space, pairwise commuting endomorphisms, and . Show that iff for all .
5. Let a finite-dimensional -vector space. Show that there with no nontrivial proper invariant subspaces.
Past exam problems
6. (6.10.2) Let be a field. Show that the only two-sided ideals of are 0 and .
7. (7.7.7) Let . If for some , show that .
8. (7.7.12) Let . Is similar to ?
9. (7.4.31) Let be the vector space of polynomials in one variable over of degree , and let be differentiation.
i) Show that .
ii) Find all eigenvectors of and .
10. (7.5.21) If , and , show that the eigenvalues of are precisely , where is an eigenvalue of .
1. Let be a field, char . For , show that there is a basis of (the ring of matrices over ) consisting only of non-diagonalizable matrices.
2. Let be a field, and a multiplicative function, i.e. for all .
i) Show that the following are equivalent:
a)
b)
c) vanishes on a proper nonempty subset of .
ii) If the conditions in (i) are satisfied, show that the vanishing set of is precisely the set of singular matrices, i.e. matrices with determinant 0 . Deduce that the set of maps satisfying (i) is in bijection with group homomorphisms from to .
3. i) Let be an endomorphism of a finite-dimensional -vector space. Suppose satisfy . Show that .
ii) Show that one can always satisfy the hypotheses of (i) by taking for some , .
4. Let be a vector space, pairwise commuting endomorphisms, and . Show that iff for all .
5. Let a finite-dimensional -vector space. Show that there with no nontrivial proper invariant subspaces.
Past exam problems
6. (6.10.2) Let be a field. Show that the only two-sided ideals of are 0 and .
7. (7.7.7) Let . If for some , show that .
8. (7.7.12) Let . Is similar to ?
9. (7.4.31) Let be the vector space of polynomials in one variable over of degree , and let be differentiation.
i) Show that .
ii) Find all eigenvectors of and .
10. (7.5.21) If , and , show that the eigenvalues of are precisely , where is an eigenvalue of .