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STAT424
Spring 2010
Homework #1
Jan 26, 2010
Homework 1
Due: Tuesday, Feb 2, 2010
1)
Questions involve the following matrices:
A
=
1 1 0 0
1 1 0 0
0 1 1 1
0 1 0 1
;
B
=
1 2 0
1 2 0
0 1 0
0 1 1
;
D
=
1 2
1 2
2 5
0 0
;
E
=
1 0
1 0
3 0
0 2
.
(a) Which of the matrices have linearly independent columns?
(b) Which of the following equalities are valid:
C
(
A
) =
C
(
B
),
C
(
A
) =
C
(
A,D
),
C
(
A,B
) =
C
(
D,E
)?
(c) Give the ranks of the matrices.
(d) Give an orthonormal basis for
C
(
A
).
(e) Find
C
(
A
)
⊥
and
C
(
D
)
⊥
(with respect to
R
4
).
2)
Write the following models in matrix notation and ﬁnd the dimension of
C
(
X
).
(a) Oneway ANOVA model
y
ij
=
μ
+
α
i
+
±
ij
,
where
i
= 1 : 3, and
j
= 1 : 2 when
i
= 1
,
2 and
j
= 1 : 3 when
i
= 3.
(b) Same as (a) but with a constraint that
α
1
=
α
2
.
(c) Same as (a) but with a consraint that
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This note was uploaded on 10/24/2010 for the course STAT 424 taught by Professor Liang,f during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Liang,F
 Variance

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