Test Bank - Linear Algebra: A Modern Introduction, 5th Edition – David Poole

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Linear Algebra
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A Modern Introduction
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5th Edition
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TEST BANK
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David Poole
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Comprehensive Test Bank for Instructors
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and Students
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9780357883495

© David Poole. All rights reserved. Reproduction or distribution
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without permission is prohibited.
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, Test Bank For
Linear Algebra A Modern Introduction 5th Edition by David Poole Copyright 2026
Section 1.0 - 1.4
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1. If u • v = 0, then ||u + v|| = ||u – v||.
a. True
b. False
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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.
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a. True
b. False

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4. The distance between two points in located by the vectors u and v is ||u – v||.
a. True
b. False
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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
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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.
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a. True
b. False


8. The area of the parallelogram with sides a, b, is || ||
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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|| .
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a. True
b. False

10. For all vectors v and scalars c, ||cv|| = c||v||.
a. True
b. False
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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.
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a. True
b. False

13. The vectors [1, 2, 3] and [k, 2k, 3k] have the same direction for all nonzero real numbers k?
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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.
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a. True
b. False

15. The distance between the planes n • x = d1 and n • x = d2 is |d1 – d2|.
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a. True
b. False
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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.
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a. True
b. False
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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.
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a. 2
b. 4
c. 6
d. 8
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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
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(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,
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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 +
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2z = –17?

28. In a parallelogram ABCD let = a, b. Let M be the point of intersection of the diagonals. Express ,
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and as linear combinations of a and b.

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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
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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
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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.
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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.

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34. Find the vector parametric equation of the line in that is perpendicular to the plane 2x – 3y + 7z – 4 = 0 and which
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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
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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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