Math 222, Fall 2004: Final Exam
•
Closed books, no notes, no calculators.
•
Upper case letters denote matrices, boldface lower case letter vectors,
and plain lower case letters scalars.
•
The transpose of a matrix
A
is denoted
A
t
.
•
The identity matrix is denoted
I
.
•
A matrix
U
is said to be orthogonal if
U
t
U
=
I
.
•
In questions 2 – 6, please provide brief explanations for the steps in
your calculations.
Question 1:
There is no connection between the matrices and the vectors
in the following questions. No motivation is required. (2p each)
(a) Suppose that
A
is an
m
×
n
matrix of rank
k
. What is dim(Col(
A
))?
(b) Suppose that there exists a nonzero vector
y
such that
A
y
= 0.
Does there exist a vector
b
such that the equation
A
x
=
b
has a
unique solution?
(c) Is it true that
λ
is an eigenvalue of
A
if and only if det(
A

λI
) = 0?
(d) Is it true that all square matrices are diagonalizable?
(e) What conditions must a matrix satisfy for it to have a singular value
decomposition?
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 Spring '07
 PeterSchultheiss
 Math, Linear Algebra, Algebra, Determinant, Vectors, Matrices, Scalar, Rn of dimension

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