STAT 8260 — Theory of Linear Models
Homework 1 – Due Tuesday, Jan. 29
Homework Guidelines:
•
Homework is due by 4:30 on the due date speciﬁed above. You may turn it in at
the beginning of class or place it in my mailbox in the Statistics Building.
No
late homeworks will be accepted without permission granted prior to
the due date.
•
Use only standard (8
.
5
×
11 inch) paper and use only one side of each sheet.
•
Homework should show enough detail so that the reader can clearly understand
the procedures of the solutions. This is
absolutely essential
for you to receive
full credit for your answer since the answers to most of the problems in Rencher
appear in the back of the book.
•
Problems should appear in the order that they were assigned.
Assignment:
1. Consider the following vectors in
R
5
:
x
= (2
,
1
,
1
,
1
,
4)
T
, and
y
= (3
,
6
,
1
,
5
,
7)
T
.
a. Find
h
x
,
y
i
,

x

2
,

y

2
, ˆ
y
=
p
(
y

x
), and
y

ˆ
y
. Show that
x
⊥
(
y

ˆ
y
),
and

y

2
=

ˆ
y

2
+

y

ˆ
y

2
.
b. Let
w
= (

1
,
2
,
4
,

4
,
0)
T
and
z
= 3
x
+ 2
w
. Show that
h
w
,
x
i
= 0 and
that

z

2
= 9

x

2
+ 4

w

2
. (Why must this be true?)
c. Let
A
1
=
{
1
,
2
}
,A
2
=
{
3
,
4
}
,A
3
=
{
5
}
. Find
p
(
y

i
A
i
),
i
= 1
,
2
,
3.
2. Is projection a linear transformation in the sense that
p
(
c
y

x
) =
cp
(
y

x
) for any
real number
c
? Prove or disprove. What is the relationship between
p
(
y

x
) and
p
(
y

c
x
) for
c
6
= 0?
3. Suppose