McGill University August 2012 Faculty of Science Supplemental Final examination

Abstract Algebra II

Math 251

???day, August ??, 2012 Time: ?:00 am to ??:00 pm

Examiner: Prof. H. Darmon Associate Examiner: Prof. E.Z.Goren

INSTRUCTIONS

1. Please write your answers \underline{clearly} in the exam booklets provided.

2. You may quote any result/theorem seen in the lectures or in the assignments without proving it (unless, of course, it is what the question asks you to prove).

3. This is a closed book exam.

4. Translation dictionary is permitted.

5. Calculators are not permitted.

This exam comprises the cover page and two pages of questions, numbered 1 to 10. Each question is worth 10 points.

Math 251 Final Exam  Page 2  August, 2012

1. a) Let F be a field, and let V be the set of m × n matrices with entries in F. Explain (briefly) why V is a vector space over F, and compute its dimension by exhibiting an explicit basis for V.

b) Suppose that V is the set of all (infinite) sequences with values in F, having only finitely many non-zero values. (I.e., a typical element of V is a sequence (a_{1}, a_{2}, …, a_{n}, …) with a_{i} ∈ F and such that a_{i} = 0 for all i ≥ M for a suitable M depending in general on the sequence (a_{i}).) Does V have a countable basis? Justify your answer.

2. A generalised Fibonnaci sequence is a sequence (a_{0}, a_{1}, …, a_{n}, …) of real numbers satisfying the recursive relation

a_{n+1} = a_{n+1} + a_{n}, for all n ≥ 0.

Show that the set of such sequences is a vector space over the field R of real numbers, and compute its dimension.

3. a) Let f(x) be a polynomial with coefficients in a field F, and let V = F [x]/(f(x)) be the quotient of the polynomial ring F [x] by the ideal generated by f(x), viewed as a vector space over F. Let T : V ⟶ V be the linear transformation given by T (p(x)) = (x + 1)p(x).

a) Compute the minimal polynomial of T.

b) Compute the characteristic polynomial of T.

c) Give a necessary and sufficient condition on f(x) for T to be diagonalisable over F.

d) Give a necessary and sufficient conditon on f(x) for T to be invertible.

4. A projection is a linear transformation (on a finite-dimensional vector space) satisfying the relation T^{2} = T. Show that a projection is always diagonalisable.

5. Let V be the real vector space of all functions of the form g(x)e^{x}, where g is a real polynomial of degree ≤ 3, and let T : V ⟶ V be the linear transformation given by

T (f) = f + 2f′.

Write down a basis for V and the matrix of T relative to this basis.

6. Define the following terms:

a) The dual space V^{∗} of a vector space V;

b) The dual linear map T^{∗} attached to a linear transformation T : V ⟶ W. Be sure to specify what the domain and target of T^{∗} are, and to write down the formula defining T^{∗}.

c) Show that (T_{1}T_{2})^{∗} = T_{2}^{∗}T_{1}^{∗} for all T_{1} : V ⟶ W and T_{2} : U ⟶ V.

7. Let V be the vector space F^{7} of 7-tuples of elements of a field F, and let W be the kernel of the map M : F^{7} ⟶ F^{3} described by the following 3 × 7 matrix (relative to the standard bases of course)

M = ( \begin{matrix}0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} ) ,

Show that two vectors in V having all but exactly one coordinate in common cannot both belong to W.

8. Let R^{3} be equipped with the standard inner product. Starting with the basis (1, 1, 0), (2, 2, 3), (3, −1, 2) of R^{3}, compute the orthonormal basis that is derived from it by performing the Gramm-Schmidt orthogonalisation process.

9. A linear transformation T on a finite-dimensional Hermitan inner product space is said to be self-adjoint if it satisfies the relation T^{∗} = T.

a) If v_{1} and v_{2} are eigenvectors for T associated to distinct eigenvalues, show that they are orthogonal.

b) Show that a self-adjoint T has an eigenvalue, and that all its eigenvalues are real.

10. Let V = R^{3} equipped with the standard dot product and resulting distance function, and let W be the hyperplane (i.e., subspace of dimension 2) defined by the equation

W = {(x, y, z) with x + y = 2z}.

Compute the vector in W which is closest to the vector (1, 1, 1), relative to the standard Euclidean distance.