1.
a) State the Lagrange Theorem explaining any terms you use.
b) Let alpha, a member of S_11, be the permutation given by
alpha(1) = 7, alpha(2) = 5, alpha(3) = 1, alpha(4) = 2, alpha(5) = 8,
alpha(6) = 9, alpha(7) = 10, alpha(8) = 4, alpha(9) = 11, alpha(10) = 3, alpha(11) = 6.
Decompose the permutation alpha first as a product of disjoint cycles and then as a product of transpsitions. What are the order and sign of alpha and alpha^-1?
c) Show that if H and K are subgroups of a group G, then the intersect of H and K is also a subgroup of G. Show that if H and K have orders 9 and 8, respectively, then the intersect of H and K contains only one element.
2. State the First Isomorphism Theorem explaining any terms you use.
b) Let G = C* be the multiplicative group of nonzero complex numbers. Is the map f: G --> G a homomorphism, provided f is given by i) f(z) = iz, ii) f(z) = z^2, iii) f(z) = |z|, iv) f(z) = z-bar? Justify your answer.
c) Let K be a field and let G be the set of all matrices of the form
a b
0 c
where a, b, c is a member of K and a =/ 0, c =/ 0. Prove that G is a group under matrix multiplication. Prove that the map g: G --> K* x K* defined by
g * the matrix: a b = (a, c)
0 c
is a homomorphism. Here K* is the set of all non-zero elements of K, considered as a multiplicative group. Prove that the kernel of g is isomorphic to the additive group of the filed K. Deduce that the set H consisting of matrices of the form
1 b
0 1
is a normal subgroup of G and the quotient group G/H is isomorphic to K* xx K*. State clearly all results that you used.
3. a) Let V be a vector space over a field K and let f: V --> V be a linear map. Suppose v is a non-zero element of V and lambda is a member of K. Explain what it means to say that v is an eigenvector of f with eigenvalue lambda. Prove that V has a basis consisting of eigenvectors of f if and only if it has a basis with respect to which the matrix representing f is diagonal.
b) Let f: R --> R be a linear map given by
f * binomial (x y) = 1 -2 * (x y)
3 -1
Find the matrix A that corresponds to the mapping f in the basis
u_1 = (1 1), u_2 = (0 1).
c) Find the characteristic and minimal polynomials of the linear map f: R^3 --> R^3 given by the matrix
2 0 0
B = 1 0 1
1 -2 3
Is there a basis for R^3 for which the matrix of f is diagonal? Justify your answer.
4. a) Let V be an inner product spave and let g: V --> V be a map. Explain what it means to say that g is an isometry. Define what it means for a square matrix to be orthogonal. Prove that the product of two orthogonal matrices is orthogonal. Explain the relationship between orthogonal matrices and isometries.
b) Find a, b and c such that the matrix
1/3 0 a
2/3 1/sqrt(2) b
2/3 -1/sqrt(2) c
is orthogonal. Does this condition determine a, b and c uniquely?
c) Let V be a subspace of R defined by
V = {(x_1, x_2, x_3, x_4) is a member of R^4 | x_1 - x_2 + x_3 - x_4 = 0}
Find an orthonormal basis of V.