Math-260 (A. Farhat)	Major Quiz No. 2		Summer, 2001
NAME:- _________________	St. No.:- _________	Sec. No.:-_____	Aug., 11, 2001

Answer all questions. Show all your work.

1. Let be the set of all polynomials of degree less than or equal to 2. Show that the set of monomials form a basis for .

2. Let be the set of all real-valued functions defined on the real line. Give an example of a subspace of .

3. Show whether the functions are linearly dependent or linearly independent.

4. Let . Show that V is a subspace of .

5. Let . Show whether or not S is a subspace of .

6. Show whether the vector is in the span .

7. For which values of a, b, and c are the vectors and linearly independent?

8. Transform the system of differential equations to a system of first-order differential equations
(x and y are functions for t)

9. Given the differential equation ,

a) Find the roots of the characteristic polynomial,

b) Give two real-valued solutions,

c) Show that the two solutions are linearly independent,

d) Give the general solution,

e) Find a particular solution satisfying and ,

f) Write the differential equation as a system of first order differential equation. Use the substitution ,

g) Find the eigenvalues of the coefficient matrix of the system. Write your observation about the relationship between the eigenvalues and the roots of the characteristic polynomial of the scalar differential equation in part (a),

h) Let , where is the solution of the differential equation in part (b) involving the cosine function. Show that y is a solution of the system of differential equations in part (f).