Test Bank For
Linear Algebra A Modern Introduction 5th Edition by David Poole Copyright 2026
Section 1.0 - 1.4
1. If u • v = 0, then ||u + v|| = ||u – v||.
a. True
b. False
2. If u • v = u • w, then either u = 0 or v = w.
a. True
b. False
3. a • b × c = 0 if and only if the vectors a, b, c are coplanar.
a. True
b. False
n
4. The distance between two points in located by the vectors u and v is ||u – v||.
a. True
b. False
5. If v is any nonzero vector, then 6v is a vector in the same direction as v with a length of 6 units.
a. True
b. False
6. The only real number c for which [c, –2, 1] is orthogonal to [2c, c, –4] is c = 2.
a. True
b. False
7. The projection of a vector v onto a vector u is undefined if v = 0.
a. True
b. False
8. The area of the parallelogram with sides a, b, is || ||
a. True
b. False
2 2 2 2
9. If a, b, c are mutually orthogonal vectors in , then (a × b • c) = ||a|| ||b|| ||c|| .
a. True
b. False
10. For all vectors v and scalars c, ||cv|| = c||v||.
a. True
b. False
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, n
11. For all vectors u, v, w in , u – (v – w) = u + w – v.
a. True
b. False
12. The projection of a vector v onto a vector u is undefined if u = 0.
a. True
b. False
13. The vectors [1, 2, 3] and [k, 2k, 3k] have the same direction for all nonzero real numbers k?
a. True
b. False
14. If a parity check code is used in the transmission of a message consisting of a binary vector, then the total number of
1’s in the message will be even.
a. True
b. False
15. The distance between the planes n • x = d1 and n • x = d2 is |d1 – d2|.
a. True
b. False
16. The zero vector is orthogonal to every vector except itself.
a. True
b. False
17. The products a × (b × c) and (a × b) × c are equal if and only if b = 0.
a. True
b. False
18. Simplify the following vector expression: 4u – 2(v + 3w) + 6(w u).
19. Find all solutions of 3x + 5 = 2 in , or show that there are no solutions.
a. 2
b. 4
c. 6
d. 8
20. Find the distance between the parallel lines.
and
21. Find the acute angle between the planes 3 and .
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,22. Find the distance between the planes and .
23. Find values of the scalar k for which the following vectors are orthogonal.
u = [k, k, –2], v = [–2, k – 1, 5]
24. Simplify the following expressions:
(a) (a + b + c) × c + (a + b + c) × b + (b – c) × a
(b) (v + 2w) ∙ (w + z) × (3z + v)
25. Find the check digit that should be appended to the vector u = [2, 5, 6, 4, 5] in if the check vector is c = [1, 1, 1, 1,
1, 1].
26. If u is orthogonal to v, then which of the following is also orthogonal to v?
27. What is the distance of the point P = (2, 3, –1) to the line of intersection of the planes 2x – 2y + z = –3 and 3x – 2y +
2z = –17?
28. In a parallelogram ABCD let = a, b. Let M be the point of intersection of the diagonals. Express ,
and as linear combinations of a and b.
2
29. Suppose that the dot product of u = [u1, u2] and v = [v1, v2] in were defined as u · v = 5u1 v1 + 2u2 v2. Consider
the following statements for vectors u, v, w, and all scalars c.
a. u · v = v · u
b. u · (v + w) = u · v + u · w
c. (cu) · v = c(u · v)
d. u · u ≥ 0 and u · u = 0 if and inly if u = 0
30. Find a value of k so that the angle between the line 4x + ky = 20 and the line 2x – 3y = –6 is 45°.
31. Find the orthogonal projection of v = [–1, 2, 1] onto the xz-plane.
32. Show that the quadrilateral with vertices A = (–3, 5, 6), B = (1, –5, 7), C = (8, –3, –1) and D = (4, 7, –2) is a square.
33. If a = [1, –2, 3], b = [4, 0, 1], c = [2, 1, –3], compute 2a – 3b + 4c.
3
34. Find the vector parametric equation of the line in that is perpendicular to the plane 2x – 3y + 7z – 4 = 0 and which
passes through the point P = (l, –5, 7).
35. Find all values of k such that d(a, b) = 6, where a = [2, k, 1, –4] and b = [3, –1, 6, –3].
36. Show that if a vector v is orthogonal to two noncollinear vectors in a plane P, then v is orthogonal to every vector in
P.
37. Final all solutions of 7x = 1 in , or show that there are no solutions.
38. Let u1 and u2 be unit vectors, and let the angle between them be radians. What is the area of the parallelogram
whose diagonals are d1 = 2u1 – u2 and d2 = 4u1 –5u2?
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,39. Solve for the vector x in terms of a and b: 2x – a = 3b = 2(a + b) – (x – b).
40. Given p = [1, –2, 1], q = [4, –4, 7], find:
a. p · q
b. ||p - q||
c. projqp
d. the cosine of the angle between p and q
3
41. Find a unit vector in in the opposite direction to v = [1, 2, –2].
42. Let ABCDEF be a regular hexagon whose sides are of length 5. If = a and = b, find the projection of
along a.
43. Find a vector of length that is orthogonal to both a = [2, 1, –3] and b = [1, –2, 1].
3
44. Suppose that the dot product of two vectors u and v in were defined as the product of the lengths of the vectors.
Which (if any) of the following statements would be true for all vectors u, v, w, and all scalars c?
a. u·v = v·u
b. u· (v + w) = u·v + u·w
c. c(u) ·v = c(u·v)
d. u·u ≥ 0 and u·u = 0 if and only if u = 0
Indicate the answer choice that best completes the statement or answers the question.
2
45. Find the normal form of the equation of the line passing through point (2,3) and having a slope of 4 in ?
a.
b.
c.
d.
46. A set of forces , , acts on an object. What is the resultant force (in vector
form)?
a.
b.
c.
d.
47. The vector form of the equation of the line in through P = (2, 0, –3) and parallel to the line with parametric
equations:
is
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, a.
b.
c.
d.
48. The equation of the plane that is equidistant from the two planes 5x – 3y + z = –3 and 10x – 6y + 2z = –7 is:
a.
b.
c.
d.
49. Which of the following vectors x is collinear with a = [2, 1, –1] and satisfies the condition a ∙ x = 3?
a.
b. [1, 1, 0]
c.
d.
50. The length of the projection of vector u onto vector v is:
a. less than or equal to length of u
b. greater than or equal to length of u
c. equal to length of v
d. equal to length of u
2
51. Which of the following equations is in the general form of a line in ?
a.
b.
c.
d.
52. Given the vectors a = [–1, 2, 1], b = [3, 1, 1], c = [4, 3, 0], the orthogonal projection of 2a – 3b along c is:
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, a.
b.
c. [–164, –123, 0]
d. [11, –1,1]
53. Which of the following expresses the fact that the vectors u and v have the same length?
a. ||u + v|| = ||u|| – ||v||
b. ||u + v|| = ||u|| + ||v||
c. u ∙ u = v ∙ v
d.
54. Given two data sets X = {1,2,3,4} and Y = {2,3,5,7}, what is the value of the Pearson correlation coefficient (r)?
a. 0.94
b. 0.96
c. 0.98
d. 0.99
2
55. Find the vector form of the equation of the line passing through points and ( ) in ?
a.
b.
c.
d.
56. A researcher finds a correlation coefficient of 0.7 between the amount of time spent studying and exam scores. Which
of the following conclusions can be drawn from this result?
a. Studying longer causes higher exam scores.
b. Higher exam scores cause longer study sessions.
c. There is a strong linear relationship between studying and exam scores.
d. 70% of exam scores can be predicted by the amount of time spent studying.
2
57. Find the parametric equation of the line passing through point (2,3) and having direction vector = [4, 5] in ?
a.
b.
c.
d.
58. A certain check digit code uses the check vector c = [9, 8, 7, 6, 5, 4, 3, 2, 1] and appends a ninth digit to an eight-digit
number u = [u1, u2, . . ., u8] so that u ∙ c = 0 in . If the first eight digits of a valid number in this scheme are 1372423,
what is the ninth digit?
a. 4
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Linear Algebra A Modern Introduction 5th Edition by David Poole Copyright 2026
Section 1.0 - 1.4
1. If u • v = 0, then ||u + v|| = ||u – v||.
a. True
b. False
2. If u • v = u • w, then either u = 0 or v = w.
a. True
b. False
3. a • b × c = 0 if and only if the vectors a, b, c are coplanar.
a. True
b. False
n
4. The distance between two points in located by the vectors u and v is ||u – v||.
a. True
b. False
5. If v is any nonzero vector, then 6v is a vector in the same direction as v with a length of 6 units.
a. True
b. False
6. The only real number c for which [c, –2, 1] is orthogonal to [2c, c, –4] is c = 2.
a. True
b. False
7. The projection of a vector v onto a vector u is undefined if v = 0.
a. True
b. False
8. The area of the parallelogram with sides a, b, is || ||
a. True
b. False
2 2 2 2
9. If a, b, c are mutually orthogonal vectors in , then (a × b • c) = ||a|| ||b|| ||c|| .
a. True
b. False
10. For all vectors v and scalars c, ||cv|| = c||v||.
a. True
b. False
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, n
11. For all vectors u, v, w in , u – (v – w) = u + w – v.
a. True
b. False
12. The projection of a vector v onto a vector u is undefined if u = 0.
a. True
b. False
13. The vectors [1, 2, 3] and [k, 2k, 3k] have the same direction for all nonzero real numbers k?
a. True
b. False
14. If a parity check code is used in the transmission of a message consisting of a binary vector, then the total number of
1’s in the message will be even.
a. True
b. False
15. The distance between the planes n • x = d1 and n • x = d2 is |d1 – d2|.
a. True
b. False
16. The zero vector is orthogonal to every vector except itself.
a. True
b. False
17. The products a × (b × c) and (a × b) × c are equal if and only if b = 0.
a. True
b. False
18. Simplify the following vector expression: 4u – 2(v + 3w) + 6(w u).
19. Find all solutions of 3x + 5 = 2 in , or show that there are no solutions.
a. 2
b. 4
c. 6
d. 8
20. Find the distance between the parallel lines.
and
21. Find the acute angle between the planes 3 and .
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,22. Find the distance between the planes and .
23. Find values of the scalar k for which the following vectors are orthogonal.
u = [k, k, –2], v = [–2, k – 1, 5]
24. Simplify the following expressions:
(a) (a + b + c) × c + (a + b + c) × b + (b – c) × a
(b) (v + 2w) ∙ (w + z) × (3z + v)
25. Find the check digit that should be appended to the vector u = [2, 5, 6, 4, 5] in if the check vector is c = [1, 1, 1, 1,
1, 1].
26. If u is orthogonal to v, then which of the following is also orthogonal to v?
27. What is the distance of the point P = (2, 3, –1) to the line of intersection of the planes 2x – 2y + z = –3 and 3x – 2y +
2z = –17?
28. In a parallelogram ABCD let = a, b. Let M be the point of intersection of the diagonals. Express ,
and as linear combinations of a and b.
2
29. Suppose that the dot product of u = [u1, u2] and v = [v1, v2] in were defined as u · v = 5u1 v1 + 2u2 v2. Consider
the following statements for vectors u, v, w, and all scalars c.
a. u · v = v · u
b. u · (v + w) = u · v + u · w
c. (cu) · v = c(u · v)
d. u · u ≥ 0 and u · u = 0 if and inly if u = 0
30. Find a value of k so that the angle between the line 4x + ky = 20 and the line 2x – 3y = –6 is 45°.
31. Find the orthogonal projection of v = [–1, 2, 1] onto the xz-plane.
32. Show that the quadrilateral with vertices A = (–3, 5, 6), B = (1, –5, 7), C = (8, –3, –1) and D = (4, 7, –2) is a square.
33. If a = [1, –2, 3], b = [4, 0, 1], c = [2, 1, –3], compute 2a – 3b + 4c.
3
34. Find the vector parametric equation of the line in that is perpendicular to the plane 2x – 3y + 7z – 4 = 0 and which
passes through the point P = (l, –5, 7).
35. Find all values of k such that d(a, b) = 6, where a = [2, k, 1, –4] and b = [3, –1, 6, –3].
36. Show that if a vector v is orthogonal to two noncollinear vectors in a plane P, then v is orthogonal to every vector in
P.
37. Final all solutions of 7x = 1 in , or show that there are no solutions.
38. Let u1 and u2 be unit vectors, and let the angle between them be radians. What is the area of the parallelogram
whose diagonals are d1 = 2u1 – u2 and d2 = 4u1 –5u2?
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,39. Solve for the vector x in terms of a and b: 2x – a = 3b = 2(a + b) – (x – b).
40. Given p = [1, –2, 1], q = [4, –4, 7], find:
a. p · q
b. ||p - q||
c. projqp
d. the cosine of the angle between p and q
3
41. Find a unit vector in in the opposite direction to v = [1, 2, –2].
42. Let ABCDEF be a regular hexagon whose sides are of length 5. If = a and = b, find the projection of
along a.
43. Find a vector of length that is orthogonal to both a = [2, 1, –3] and b = [1, –2, 1].
3
44. Suppose that the dot product of two vectors u and v in were defined as the product of the lengths of the vectors.
Which (if any) of the following statements would be true for all vectors u, v, w, and all scalars c?
a. u·v = v·u
b. u· (v + w) = u·v + u·w
c. c(u) ·v = c(u·v)
d. u·u ≥ 0 and u·u = 0 if and only if u = 0
Indicate the answer choice that best completes the statement or answers the question.
2
45. Find the normal form of the equation of the line passing through point (2,3) and having a slope of 4 in ?
a.
b.
c.
d.
46. A set of forces , , acts on an object. What is the resultant force (in vector
form)?
a.
b.
c.
d.
47. The vector form of the equation of the line in through P = (2, 0, –3) and parallel to the line with parametric
equations:
is
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, a.
b.
c.
d.
48. The equation of the plane that is equidistant from the two planes 5x – 3y + z = –3 and 10x – 6y + 2z = –7 is:
a.
b.
c.
d.
49. Which of the following vectors x is collinear with a = [2, 1, –1] and satisfies the condition a ∙ x = 3?
a.
b. [1, 1, 0]
c.
d.
50. The length of the projection of vector u onto vector v is:
a. less than or equal to length of u
b. greater than or equal to length of u
c. equal to length of v
d. equal to length of u
2
51. Which of the following equations is in the general form of a line in ?
a.
b.
c.
d.
52. Given the vectors a = [–1, 2, 1], b = [3, 1, 1], c = [4, 3, 0], the orthogonal projection of 2a – 3b along c is:
Copyright Cengage Learning. Powered by Cognero. Page 5
, a.
b.
c. [–164, –123, 0]
d. [11, –1,1]
53. Which of the following expresses the fact that the vectors u and v have the same length?
a. ||u + v|| = ||u|| – ||v||
b. ||u + v|| = ||u|| + ||v||
c. u ∙ u = v ∙ v
d.
54. Given two data sets X = {1,2,3,4} and Y = {2,3,5,7}, what is the value of the Pearson correlation coefficient (r)?
a. 0.94
b. 0.96
c. 0.98
d. 0.99
2
55. Find the vector form of the equation of the line passing through points and ( ) in ?
a.
b.
c.
d.
56. A researcher finds a correlation coefficient of 0.7 between the amount of time spent studying and exam scores. Which
of the following conclusions can be drawn from this result?
a. Studying longer causes higher exam scores.
b. Higher exam scores cause longer study sessions.
c. There is a strong linear relationship between studying and exam scores.
d. 70% of exam scores can be predicted by the amount of time spent studying.
2
57. Find the parametric equation of the line passing through point (2,3) and having direction vector = [4, 5] in ?
a.
b.
c.
d.
58. A certain check digit code uses the check vector c = [9, 8, 7, 6, 5, 4, 3, 2, 1] and appends a ninth digit to an eight-digit
number u = [u1, u2, . . ., u8] so that u ∙ c = 0 in . If the first eight digits of a valid number in this scheme are 1372423,
what is the ninth digit?
a. 4
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