COT4501 Spring 2012
Homework II Solutions
This assignment has eight problems and they are equally weighted.
The assignment
is due in class on Tuesday, Tuesday 14, 2012. There are six regular problems and two
computer problems (using MATLAB). For the computer problems, turn in your results
(e.g., graphs, plots, simple analysis and so on) and also a printout of your (MATLAB)
code.
Problem 1
Provide short answers to the following questions:
• True or false: At the solution to a linear least squares problem
Ax
’
b
, the residual
vector
r
=
b

Ax
is orthogonal to span
(
A
)
.
Solution:
True. It is an important fact that the residual vector is orthogonal to
span
(
A
)
.
• True or false: In solving a linear least squares problem
Ax
’
b
, if the vector
b
lies
in span
(
A
)
, then the residual is
0
.
Solution:
True. This means that there is a vector
x
, such that
Ax
=
b
and
r
=
Ax

b
= 0
.
• In a linear least squares problem
Ax
’
b
, where
A
is an
m
×
n
matrix, if rank
(
A
)
<
n
, then which of the following situation are possible?
1. There is no solution.
2. There is a unique solution.
3. There is a solution, but it is not unique.
Solution:
There is a solution but it is not unique. The nonuniqueness comes from
the fact that columns of
A
are linearly dependent.
• in Solving an overdetermined least squares problem
Ax
’
b
,which would be more
serious difficulty: that the rows of
A
are linearly dependent, or that the columns
of A are linear dependent? Explain.
Solution:
If columns of
A
are (nearly) linear dependent, the matrix
A
>
A
becomes
(nearly) singular with a large condition number. This implies that the problem
Ax
’
b
could be unstable with respect to small perturbation of
b
. Dependent rows
of
A
typically do not cause serious problems.
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 Spring '08
 Davis
 Linear Algebra, Vector Space, Linear least squares, Ax B

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