MASTERS COMPREHENSIVE EXAM ADVANCED LINEAR ALGEBRA AUGUST 1999

(1) Let W be the intersection of the three ‘hyperplanes’

3x_{2} + x_{3} + 2x_{4} − x_{5} − x_{6} + 4x_{7} = 0,
x_{1}+3x_{2} + x_{3} + x_{4} + 2x_{5} − x_{6} − 4x_{7} = 0,
x_{1}−6x_{2} − 2x_{3} − 2x_{4} + 5x_{5} + 2x_{6} − 16x_{7} = 0,

in ℝ^{7}. What is the dimension of W? Explain your answer.

(2) Let V be a finite dimensional vector space with an inner product (denoted by ( , )). Fix two vectors w_{1}, w_{2} that are orthogonal and of length one. Define a function T : V → V by

T (v) = (v, w_{1})w_{1} + (v, w_{2})w_{2}.

(a) Show that T is a linear transformation.

(b) Find T (w_{1}), T (w_{2}) and T (w), for any w orthogonal to both w_{1} and w_{2}.

(c) Describe T geometrically.

(d) Determine the minimal polynomial of T.

(e) Determine the Jordan form of T.

(f) What is the rank of T?

(3) Suppose T : ℝ^{5} → ℝ^{5} is a linear transformation with characteristic polynomial (x − 3)^{5}. (a) Find the determinant of T.

(b) Can you find two linearly independent vectors v_{1}, v_{2} ∈ ℝ^{5} so that T (v_{1}) and T (v_{2}) are linearly dependent? Explain your answer.

(4) Let S, T be linear transformations of a vector space V. Suppose that S is invertible and T S = ST^{4}. Show that if v is an eigenvector for T then S(v) is also an eigenvector for T.