Math 2270 Practice Exam 3
-
University of Utah
Fall 2012
Name:
I€
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Please show all your work, as a worked
This is a 50 minute exam. 7
problem is required for full points, and partial credit may be rewarded for
some work in the right direction.
1
1. (9 Points) The Four Subspaces
• (3 points) The rowspace is the orthogonal complement of the
• (3 points) I an n-i x n matrix has a d-dimensional nullspace, then
its rowspace has dimension
cI
—
• (3 points) For an m x n matrix A calculate:
dim(C(A)) + dirn(N(A
)).
T
11
2
2. (8 points) Orthogonal Coinplernents
complement of the vector space
3
-
Find a basis for the orthogonal
3. (20 points) Projections Find the projection of the vector b onto the
column space of the matrix A:
-
/ 4 \
—1 I
b=( 01I.
1 0 i\
1 1 1
A=(
011
110)
1
‘\1)
(iIot\
t
A
/
(0]
i 0
0 H
1
/
3)
I
Z
3Z
Zl3
10)
Y()
x1
)
L3ZO
3
\j
C
O3
I
(z
33
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3
‘5
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f
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2
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L 31
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( 10 I
oti)
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(
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{
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oI
l
ijH
31
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4. (15 points) Linear Regression Find the least squares regression line
through the points (1, 1), (2, 3), and (4. 5).
-
iH
1
,
/
5
A4/I I
()
At;
43
7
1/3 7
L
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Y
I
3+7y
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7)((IYL
5
7
7
5. (15 points) Orthonormal Bases Find an orthonormal basis for the vec
tor space
-
/
2 \
\_2)
0
span{/1\
2),(
\2)J
p
Ii
((70)
33D
10
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lol
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1
i)
H)
(0)
(1zo)/
11
-
(lLo)
(S
z
[)
(o}
(z)- (1*
(-0
/0)
- H)(JI
-
5
6. (8 points) Properties of Matrices Using the three fundamental prop
erties of the determinant (determinant of the identity is 1, switching
rows switches the sign, linearity for each row) prove that a matrix
with a repeated row has determinant 0.
-
A
6- 1IflQ4
y’
-ef(A)
7
7. (15 points) Calculating Determinants
the matrix
(
-
Calculate the determinant of
1 —2 7 9 \
3 —4 5 5
3 6 1 —1 I
4532)
Ye
1-Z7
O
/
/J77
/
(iZ
-
C C 7C iO
00771cc! )-7
C
—
IYZ7K
icj
(ii)
8
/)(2 /7
8. (10 points) Other Ways of Calculating Determinants Calculate the de
terminant of the matrix
-
/4 0 2
(231
1 5
using a cofactor expansion along column 2.
3
j
C
-
9