Math-260-021 (A. Farhat) Major Exam  2 Dec., 30, 2002 NAME:- _________________ St. #: ____________ Sec. #: _____ List #: ______

1. Given the linear system

a)      Use Cramer’s Rule to solve the system

b)      Find the inverse of the coefficient matrix of the system using the adjoint method.

c)      Check your answer in part (a) by solving the system using the inverse method.

2.      Express the vector  as a linear combination of the vectors  and .

3.      Show whether or not  W = {  |  z = 2 x +3 y } is a subspace of .

4.      Given the linear system

a)      Find a basis for the solution space of the system.

b)      Show that the bases vectors are linearly independent.

c)      Give the dimension of the solution space.

d)      Is the set of bases vectors a spanning set of R4. Why?

5.      Given the initial value problem  y'' + 4 y = 2 cos(2 x),   y(0) = ,  y'(0) = 4

a)      find the complementary solution

b)      use the method of undetermined coefficients to find a particular solution

c)      find the general solution

d)      find a particular solution satisfying the initial conditions

1. If  -1 is a root of the characteristic equation of the differential equation

find the form of the particular solution.