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Exam
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find a QR factorization of the matrix A.
1)
A =
0 1 1
1 1 0
-1-1 1
1 -1 1
1)
A)
Q =
0 3 14
1 2 2
-1-2 9
1 -4 7
, R =
330 0
1311
33 0
0-3
33 30
330
B)
Q =
0 3 14
1 2 2
-1-2 9
1 -4 7
, R =
33130
011
33 -3
33
0 0 30
330
C)
Q =
03
33 14
330
132
33 2
330
-13-2
33 9
330
13-4
33 7
330
, R =
33130
011
33 -3
33
0 0 30
330
D)
Q =
03
33 14
330
132
33 2
330
-13-2
33 9
330
13-4
33 7
330
, R =3 1 0
011 -3
0 0 30
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
2)
For f, g in C[a, b], set f, g =b
af(t) g(t) dt .
Show that f, g defines an inner product of C[a, b].
2)
3)
Let t0, ...., tn be distinct real numbers. For p and q in n, define p, q = p(t0)q(t0) +
p(t1)q(t1) + ... + p(tn)q(tn).
3)
4)
With the given positive numbers, show that vectors u = (u1, u2)and v = (v1, v2) define an
inner product in
2 using the 4 axioms.
Set u, v =2u1v1+2u2v2
4)
5)
Let V be the space C[0, 1] and let W be the subspace spanned by the polynomials p1(t) = 1,
p2(t) =1t - 1, and p3(t) = 12t2. Use the Gram-Schmidt process to find an orthogonal basis
for W.
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Compute the dot product u · v.
6)
u=-1
3
3 , v=5
2
-3
6)
A)
0
B)
8
C)
-8
D)
-2
Find the orthogonal projection of y onto u.
7)
y=-3
-4, u=5
-10
7)
A)
5
-10
B)
1
5
-2
5
C)
25
-50
D)
1
-2
Compute the dot product u · v.
8)
u=5
0, v=12
-10
8)
A)
50
B)
70
C)
60
D)
-50
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
9)
Let x1=4
-2
0, x2=10
-30
2
9)
A)
4
-2
0, 30
-40
2
B)
4
-2
0, -10
-30
-2
C)
-6
-2
0, 10
-30
2
D)
4
-2
0, -10
-20
2
Find the distance between the two vectors.
10)
u= (0, 0, 0) , v = (-8, -6, -2)
10)
A)
104
B)
-16
C)
226
D)
4-1
Determine whether the set of vectors is orthogonal.
11)
20
40
20 , -20
0
20 , 20
20
20
11)
A)
No
B)
Yes
Find a unit vector in the direction of the given vector.
12)
16
-32
12)
A)
13
-23
B)
1
5
-2
5
C)
15
25
D)
15
-25
Express the vector x as a linear combination of the u's.
13)
u1=1
-2, u2=6
3, x=8
14
13)
A)
x =2u1- 4u2
B)
x =4u1- 2u2
C)
x = -4u1+ 2u2
D)
x = -4u1- 2u2
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
14)
Let x1=
0
1
-1
1
, x2=
1
1
-1
-1
, x3=
1
0
1
1
14)
A)
0
1
-1
1
,
1
1
-1
-1
,
1
0
1
1
B)
0
1
-1
1
,
1
0
0
-2
,
6
0
1
3
C)
0
1
-1
1
,
3
2
-2
-4
,
14
2
9
7
D)
0
1
-1
1
,
3
4
-4
-2
,
18
4
19
13
Find a QR factorization of the matrix A.
15)
A =6-6
-318
0-6
15)
A)
Q =6 6
-312
0 -6, R =
15
5-30
5
0 -36
6
B)
Q =6-6
-318
0 -6, R =
-30
5 0
-36
615
5
C)
Q =
25-16
-15-26
0 16
, R =
15
5-30
5
0 -36
6
D)
Q =
25-16
-15-26
0 16
, R =3-6
0 -6
Find the least-squares line y =0+zx that best fits the given data.
16)
Given: The data points (-2, 2), (-1, 5), (0, 5), (1, 2), (2, 2).
Suppose the errors in measuring the y-values of the last two data points are greater than for the
other points. Weight these data points twice as much as the rest of the data.
X =
1-2
1-1
1 0
1 1
1 2
, =1
2, y =
2
5
5
2
2
16)
A)
y =2.6 - 0.53x
B)
y =2.8 - 0.55x
C)
y =5.8 - 0.90x
D)
y =2.9 - 0.45x
Compute the length of the given vector.
17)
p(t) =10t2 and q(t) = t - 1, where t0= 0, t1=1
2, t2= 1
17)
A)
p = 0.5 70; q =5
4
B)
p =5
4; q =10 5
4
C)
p =5
217; q =5
4
D)
p =10 17; q =5
4
Solve the problem.
18)
Let C[0, ] have the inner product f, g =
0f(t)g(t) dt, and let m and n be unequal positive
integers. Prove that cos(mt) and cos(nt) are orthogonal.
18)
A)
cos(mt), cos(nt) =
0cos(mt)cos(nt)dt
=1
2
0[cos(mt + nt) + cos(mt - nt)]dt
=1
2sin(mt + nt)
m + n +sin(mt - nt)
m - n from [0, ]
= 0.
6
B)
cos(mt), cos(nt) =
0cos(mt)cos(nt)dt
=1
2
0[cos(mt + nt) - cos(mt - nt)]dt
=1
2sin(mt + nt)
m - n -sin(mt - nt)
m+ n from [0, ]
= 0.
C)
cos(mt), cos(nt) =
0cos(mt)cos(nt)dt
=1
2
0[cos(mt + nt) + cos(mt - nt)]dt
=1
2sin(mt + nt)
m - n +sin(mt - nt)
m + n from [0, ]
= 1.
D)
cos(mt), cos(nt) =
0cos(mt)cos(nt)dt
=
0[cos(mt + nt) + cos(mt - nt)]dt
=sin(mt - nt)
m + n +sin(mt + nt)
m - n from [0, ]
= 1.
Determine whether the set of vectors is orthogonal.
19)
3
6
3, -3
0
3, 3
-3
3
19)
A)
No
B)
Yes
Compute the dot product u · v.
20)
u=1
14 , v=7
1
20)
A)
23
B)
100
C)
21
D)
7
7
Find the distance between the two vectors.
21)
u= (-8, 16), v= (16, -16)
21)
A)
8
B)
1600
C)
200
D)
40
Find a least-squares solution of the inconsistent system Ax = b.
22)
A =1 2
3 4
5 9 , b =3
2
1
22)
A)
19
18
-7
18
B)
-3
4
-1
4
C)
37
84
-5
21
D)
57
7016
-21
7016
Solve the problem.
23)
Let V be in 4, involving evaluation of polynomials at -6, -3, 0, 3, and 6, and view 2 by applying
the Gram-Schmidt process to the polynomials 1, t, and t2.
23)
A)
p2(t) =t2-18
5
B)
p2(t) =t2-72
5
C)
p2(t) =t2- 18
D)
p2(t) =t2+ 18
Find the least-squares line y =0+zx that best fits the given data.
24)
Given: The data points (-3, 2), (-2, 5), (0, 5), (2, 2), (3, 7).
Suppose the errors in measuring the y-values of the last two data points are greater than for the
other points. Weight these data points half as much as the rest of the data.
X =
1-3
1-2
1 0
1 2
1 3
, =1
2, y =
2
5
5
2
7
24)
A)
y =0.45 + 4.6x
B)
y =4.9 + 1.54x
C)
y =4.6 + 0.45x
D)
y =0.9 + 1.54x
Solve the problem.
25)
Let V be in 4, involving evaluation of polynomials at -3, -1, 0, 1, and 3, and view 2 by applying
the Gram-Schmidt process to the polynomials 1, t, and t2.
25)
A)
p2(t) =t2-5
3
B)
p2(t) =t2+16
5
C)
p2(t) =t2- 4
D)
p2(t) =t2+ 4
Let W be the subspace spanned by the u's. Write y as the sum of a vector in W and a vector orthogonal to W.
26)
y =21
3
9, u1= 1
0
-1, u2=2
1
2
26)
A)
y=15
0
-15 +6
3
6
B)
y=20
7
8+ 1
-4
1
C)
y=20
7
8+41
10
17
D)
y=20
7
8+-1
4
-1
Find the orthogonal projection of y onto u.
27)
y=8
-24 , u=8
-4
27)
A)
16
-4
B)
8
-4
C)
16
-8
D)
4
2
Find the equation y =0+1x of the least-squares line that best fits the given data points.
28)
Data points: (5, -3), (2, 2), (4, 3), (5, -1)
X =
1 5
1 2
1 4
1 5
, y =
-3
2
3
-1
28)
A)
y =67
12 - 2x
B)
y =19
2-8
3x
C)
y =67
12 + 0x
D)
y =67
12 -4
3x
Find the distance between the two vectors.
29)
u= (25, 16, 21) , v= (-3, 2, 7)
29)
A)
14 6
B)
1176
C)
2398
D)
56
Let W be the subspace spanned by the u's. Write y as the sum of a vector in W and a vector orthogonal to W.
30)
y =12
14
25 , u1= 2
2
-1, u2=-1
3
4
30)
A)
y=1
21
17 +13
35
42
B)
y=2
42
34 +10
-28
-9
C)
y=1
21
17 +-11
7
-8
D)
y=1
21
17 + 11
-7
8
10
Given A and b, determine the least-squares error in the least-squares solution of Ax = b.
31)
A =4 3
2 1
3 2 , b =4
2
1
31)
A)
3.82970843
B)
260.495468
C)
1.63299316
D)
91.929925
Solve the problem.
32)
Find the nth-order Fourier approximation to the function f(t) =4t on the interval [0, 2].
32)
A)
4-8cos(t) -4sin(2t) -8
3cos(3t) - ... -8
ncos(nt)
B)
4-8sin(t) -4sin(2t) -4
3sin(3t) - ... -4
nsin(nt)
C)
4-8sin(t) -4sin(2t) -8
3sin(3t) - ... -8
nsin(nt)
D)
- cos(t) - cos(2t) - cos(3t) - ... -8
ncos(nt)
Find the closest point to y in the subspace W spanned by u1 and u2.
33)
y =15
1
7, u1=1
0
-1, u2=2
1
2
33)
A)
20
9
16
B)
-14
-5
-6
C)
13
4
3
D)
14
5
6
Find the distance between the two vectors.
34)
u= (-8, 2, -6) , v= (-3, 3, 4)
34)
A)
-16
B)
314
C)
5 6
D)
126
35)
u= (0, 0, 0) , v= (6, 9, 9)
35)
A)
322
B)
198
C)
2 6
D)
24
Compute the dot product u · v.
36)
u=-12
4, v=0
15
36)
A)
72
B)
48
C)
-180
D)
60
37)
u=-16
0
-8, v=2
3
-1
37)
A)
-40
B)
-21
C)
0
D)
-24
Find the closest point to y in the subspace W spanned by u1 and u2.
38)
y =10
20
33 , u1= 2
2
-1, u2=-1
3
4
38)
A)
-8
96
95
B)
-1
27
25
C)
1
-27
-25
D)
11
23
5
Find the equation y =0+1x of the least-squares line that best fits the given data points.
39)
Data points: (2, 1), (3, 2), (7, 3), (8, 4)
X =
1 2
1 3
1 7
1 8
, y =
1
2
3
4
39)
A)
y =5
13 +4
13x
B)
y = - 15
13 +11
26x
C)
y =5
13 +11
26x
D)
y = - 3 +11
26x
Compute the dot product u · v.
40)
u=-15
9, v=-14
-4
40)
A)
-36
B)
210
C)
174
D)
246
Find a least-squares solution of the inconsistent system Ax = b.
41)
A =
1 1 0 0
1 1 0 0
1 0 1 0
1 0 1 0
1 0 0 1
1 0 0 1
, b =
7
8
0
2
4
1
41)
A)
5
4
5
-3
2
0
+x4
-1
1
1
0
B)
5
2
5
-7
2
0
+x4
-1
1
1
1
C)
5
2
4
-3
2
0
+x4
-1
-1
1
1
D)
5
2
5
-3
2
0
+x4
-1
1
1
1
13
Express the vector x as a linear combination of the u's.
42)
u1=-2
0
1 , u2=3
5
6, u3=-2
6
-4, x=4
-14
33
42)
A)
x =10u1+ 4u2- 8u3
B)
x = -5u1+ 4u2+ 4u3
C)
x =5u1+ 2u2- 4u3
D)
x = -5u1- 2u2+ 4u3
Find the distance between the two vectors.
43)
u= (6, -1), v= (1, 6)
43)
A)
37
B)
74
C)
37
D)
74
Compute the length of the given vector.
44)
p(t) =4t +7 and q(t) =7t - 5, where t0= 0, t1= 1, t2= 2
44)
A)
p =395; q =110
B)
p =395; q =340
C)
p =353; q =390
D)
p = -297; q = -60
Find the distance between the two vectors.
45)
u= (14, 17, 20) , v= (1, 7, 4)
45)
A)
521
B)
39
C)
525
D)
917
Find a unit vector in the direction of the given vector.
46)
-32
32
-16
46)
A)
2
3
2
3
1
3
B)
-2
3
2
3
-1
3
C)
-2
9
2
9
-1
9
D)
-25
25
-15
Answer Key
Testname: C6
16
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