Math 701 Exam 1 Version A
1. Given the lemmata
Lemma 1. The matrix I vv is unitary if and only if v
or v = 0.
2
2
=2
Lemma 2. Let x and y be two vectors such that x 2 = y 2
and x, y is real. Then there exists a unitary matrix U of the
form I vv such that
1. [Kincaid and Cheney Exercise 1.3#11a] Give bases consisting of real sequences for
the solution space of (4I 3E 2 + E 3 )x = 0.
This problem maybe reformulated as nding a basis for the solutions to
p(E )x = 0
where
p() = 4 32 + 3 .
Factor p as p() = ( +
Math 701 Final Version A
1. Answer one of the following questions:
(i) For x Rn and A Rnn prove that A
2
= (ATA)1/2 .
(ii) Prove every nonconstant polynomial has at least one root in C.
Math 701 Final Version A
2. Answer one of the following questions:
(i
1. [Kincaid and Cheney Problem 3.2#4] Steensens method is given by the iteration
f x + f ( x) f ( x)
f ( xn )
where
g ( x) =
.
g ( xn )
f ( x)
Show this method is quadratically convergent under suitable hypothesis.
xn+1 = xn
Let r be a solution such that
1. [Kincaid and Cheney Problem 5.2#1] Find the Schur factorization of
A1 =
38
2 3
and
A2 =
4
1
7
.
2
This problem can be worked by hand. For accuracy we use Maple to perform each step in
the proof on Schurs theorem. The Maple script is
1
2
# Kincaid and C
Math 701 Quiz 1 Version A
INSTRUCTIONS: Complete 3 questions out of the 6 questions below. Clearly indicate which problems you wish graded. Work each
problem on a separate sheet of paper (or more if necessary).
1. For x Rn and A Rnn dene
x2 + x2 + + x2 ,
1. [Kincaid and Cheney Problem 6.8#6] Show that the Hilbert matrix with elements
aij = (i + j + 1)1 for i, j = 0, 1, 2, . . . , n 1 is a Gram matrix for the functions
1, x, x2 , . . . , xn1 .
We dene the inner product
1
f, g =
f (x)g (x) dx
0
on the space
Math 701 Quiz 2 Version A
INSTRUCTIONS: Complete 2 questions out of the 2 questions below. Clearly indicate which problems you wish graded. Work each
problem on a separate sheet of paper (or more if necessary).
1. Let f be a function in C n+1 [a, b] and l