Qualifying Doctoral Exam:
Linear Algebra and Numerical Analysis.
Wednesday, September 2, 1998.
You have three hours for this exam. Show all working in the books provided.
1. (a) Write the di erential equation y + y = 0 as a rst order system in the form
00
du=dt = A u. Use the eigenvalues and eigenvectors of the matrix A to compute the solution of the di erential equation y + y = 0 with the initial conditions y0 = 2 and y0 = 0.
00
(b) Find the Jordan form of the matrix A given by
0 2 ;1 0
B
A = B 0 3 ;1
B0 1 1
@
0 ;1 0
0
11
0 C:
C
A
0C
3
2. (a) If V is the subspace spanned by the vectors u1, u2, and u3, where
01
01
01
1
1
1
u1 = B 1 C u2 = B 2 C u3 = B 5 C
@A
@A
@A
0
0
0
nd a matrix A that has V as its row space and a matrix B that has V as its nullspace.
(b) If A is a square matrix, show that the nullspace of A2 contains the nullspace of A. Show
also that the column space of A2 is contained in the column space of A.
3. (a) Let A be an invertible n n matrix. If A = L1D1U1 and ;1 = L2D2 U2, prove that
A
;1
L1 = L2, U1 = U2, and D1 = D2. Start by deriving the equation L1 L2D2 = D1 U1U2 .
(b) Prove that expression the for any vector in a vector space in terms of the vectors in a
basis of the space is unique.
4. Derive Simpson's rule for approximating Rab f (x)dx by using a quadratic Lagrange poly-
nomial.
(a) What is the order of the error?
(b) Using Taylor series, derive Simpson's rule with a higher order term.
5. (a) Apply Newton's method to the function
(p
f (x) = px
x0
; ;x x 0
which has the unique root = 0. What is the behavior of the iterates? Do they converge,
and if so, at what rate?
(b) Do the same as in (a) but with
( p2
3
x x0
f (x) = ;px2 x 0
3
(c) Explain your answers to (a) and (b).
6. (a) Assume that that the matrix A 2 Rn n is singular, so that the system Ax = b does
not have a unique solution. Consider the matrix splitting scheme A = B +(A ; B), where B
is nonsingular. Show that the corresponding iterative scheme xn+1 = (I ; B;1 A)xn + B;1b
does not converge.
(b) For A nonsingular, the iterative scheme will converge provided what criteria is met?
(c) Find the rst two iterations of the Jacobi method for the linear system
3x1 ; x2 + x3 = 1
3x1 + 6x2 + 2x3 = 0
3x1 + 3x2 + 7x3 = 4