Master’s Comprehensive Exam ADVANCED LINEAR ALGEBRA June 1997

1. (30 pts.) Let

A = ( \begin{matrix}1 & 2 & −1 & 3 \\ −2 & −4 & 2 & −6 \\ 0 & 0 & 1 & 5 \end{matrix} ) .

(a) Find the rank of A.

(b) Give a basis for the kernel ker(A).

(c) Give a basis for the row space Row(A).

(d) Give a basis for the column space Col(A).

(e) Give the relationship between ker(A) and Row(A).

2. (20 pts.) A matrix A has characteristic polynomial

c_{A}(x) = (x − 3)^{4}(x + 2)^{2}.

(a) State all possible minimal polynomials for A, and give examples of matrices A with characteristic polynomial c_{A}(x) and with each possible minimal polynomial.

(b) Give an example of two matrices A and B which have the same characteristic polynomial and the same minimal polynomial yet are not similar. Explain why the matrices are not similar.

3. (25 pts.) Consider a vector space V over R with basis B = {v_{1}, …, v_{n}}.

(a) Define what we mean by a bilinear form b on V.

(b) Give the matrix A = (a_{ij}) for the bilinear form b on V with respect to the basis B.

(c) Suppose that C = {w_{1}, …, w_{n}} is a new choice of basis for V and that P is the corresponding change of basis matrix. Describe the matrix for b with respect to C in terms of the old matrix A and the change of basis matrix P.

(d) Consider the bilinear form b on R^{3} given by

b(x, y) = x_{1}y_{1} + 2x_{2}y_{2} + 4x_{3}y_{3} − x_{1}y_{2} − x_{2}y_{1} + 2x_{2}y_{3} + 2x_{3}y_{2}.

Find a basis of R^{3} with respect to which the matrix of b is diagonal, and give the matrix of b with respect to your choice of basis.

(over)

4. (25 pts.) Let V be a vector space over a field F with inner product ⟨v, w⟩. Prove Bessel’s inequality that whenever v_{1}, v_{2}, …, v_{n} ∈ V are nonzero and orthogonal and v ∈ V, then

∑_{i=1}^{n} \frac{|⟨v, v_{i}⟩|^{2}}{⟨v_{i}, v_{i}⟩} ≤ ⟨v, v⟩.

(Hint: Write v = w_{1} + w_{2} with w_{1} ∈ span{v_{1}, …, v_{n}} and w_{2} ∈ span{v_{1}, …, v_{n}}^{⊥}.)