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
Answer Key