Elementary Algebra For College Students (10th Edition)
10th Edition
ISBN: 9780134759005
Author: Allen R. Angel, Dennis Runde
Publisher: PEARSON
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Chapter 3.2, Problem 75ES
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1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set
Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k
components, where k is the greatest common divisor of {n, r,s}.
Question 3
over a field K.
In this question, MË(K) denotes the set of n × n matrices
(a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is
equivalent to A-¹? Justify your answer.
(b) Let B be given by
8
B = 0 7 7
0 -7 7
Working over the field F2 with 2 elements, compute the rank of B as an element
of M2(F2).
(c) Let
1
C
-1 1
[4]
[6]
and consider C as an element of M3(Q). Determine the minimal polynomial
mc(x) and hence, or otherwise, show that C can not be diagonalised.
[7]
(d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write
down all the eigenvalues. Show your working.
[8]
R denotes the field of real numbers, Q denotes the field of rationals, and
Fp denotes the field of p elements given by integers modulo p. You may refer to general
results from lectures.
Question 1
For each non-negative integer m, let R[x]m denote the
vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m.
x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent
(a) Let vi = x, V2 =
list in R[x] 3.
(b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4)
is a basis of R[x] 3.
[8]
[6]
(c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a
linear map.
[6]
(d) Write down the matrix for the map ƒ defined in (c) with respect to the basis
(2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3.
[5]
Chapter 3 Solutions
Elementary Algebra For College Students (10th Edition)
Ch. 3.1 - Fill in the blanks with the appropriate word,...Ch. 3.1 - Prob. 2ESCh. 3.1 - Fill in the blanks with the appropriate word,...Ch. 3.1 - Prob. 4ESCh. 3.1 - Prob. 5ESCh. 3.1 - Prob. 6ESCh. 3.1 - Prob. 7ESCh. 3.1 - Prob. 8ESCh. 3.1 - Prob. 9ESCh. 3.1 - Prob. 10ES
Ch. 3.1 - Prob. 11ESCh. 3.1 - Prob. 12ESCh. 3.1 - Prob. 13ESCh. 3.1 - Prob. 14ESCh. 3.1 - Prob. 15ESCh. 3.1 - Prob. 16ESCh. 3.1 - Prob. 17ESCh. 3.1 - Prob. 18ESCh. 3.1 - Prob. 19ESCh. 3.1 - Prob. 20ESCh. 3.1 - Prob. 21ESCh. 3.1 - Prob. 22ESCh. 3.1 - Prob. 23ESCh. 3.1 - Prob. 24ESCh. 3.1 - Prob. 25ESCh. 3.1 - Prob. 26ESCh. 3.1 - Prob. 27ESCh. 3.1 - Prob. 28ESCh. 3.1 - Prob. 29ESCh. 3.1 - Prob. 30ESCh. 3.1 - Prob. 31ESCh. 3.1 - Prob. 32ESCh. 3.1 - Prob. 33ESCh. 3.1 - Prob. 34ESCh. 3.1 - Prob. 35ESCh. 3.1 - Prob. 36ESCh. 3.1 - Prob. 37ESCh. 3.1 - Prob. 38ESCh. 3.1 - Prob. 39ESCh. 3.1 - In Exercises 31-44, determine which unknown...Ch. 3.1 - Prob. 41ESCh. 3.1 - Prob. 42ESCh. 3.1 - Prob. 43ESCh. 3.1 - Prob. 44ESCh. 3.1 - Prob. 45ESCh. 3.1 - Prob. 46ESCh. 3.1 - Prob. 47ESCh. 3.1 - Prob. 48ESCh. 3.1 - Prob. 49ESCh. 3.1 - Prob. 50ESCh. 3.1 - Prob. 51ESCh. 3.1 - Prob. 52ESCh. 3.1 - Prob. 53ESCh. 3.1 - Prob. 54ESCh. 3.1 - Prob. 55ESCh. 3.1 - Prob. 56ESCh. 3.1 - Prob. 57ESCh. 3.1 - Prob. 58ESCh. 3.1 - Prob. 59ESCh. 3.1 - Prob. 60ESCh. 3.1 - Prob. 61ESCh. 3.1 - Prob. 62ESCh. 3.1 - Prob. 63ESCh. 3.1 - Prob. 64ESCh. 3.1 - Prob. 65ESCh. 3.1 - Prob. 66ESCh. 3.1 - Prob. 67ESCh. 3.1 - Prob. 68ESCh. 3.1 - Prob. 69ESCh. 3.1 - Prob. 70ESCh. 3.1 - Prob. 71ESCh. 3.1 - Prob. 72ESCh. 3.1 - Prob. 73ESCh. 3.1 - Prob. 74ESCh. 3.1 - Prob. 75ESCh. 3.1 - Prob. 76ESCh. 3.1 - Prob. 77ESCh. 3.1 - Prob. 78ESCh. 3.1 - Prob. 79ESCh. 3.1 - Prob. 80ESCh. 3.1 - Prob. 81ESCh. 3.1 - Prob. 82ESCh. 3.1 - Prob. 83ESCh. 3.1 - Prob. 84ESCh. 3.1 - Prob. 85ESCh. 3.1 - Prob. 86ESCh. 3.1 - Prob. 87ESCh. 3.1 - Prob. 88ESCh. 3.1 - Prob. 89ESCh. 3.1 - Prob. 90ESCh. 3.1 - Prob. 91ESCh. 3.1 - Prob. 92ESCh. 3.1 - Prob. 93ESCh. 3.1 - Prob. 94ESCh. 3.1 - Prob. 95ESCh. 3.1 - Prob. 96ESCh. 3.1 - Prob. 97ESCh. 3.1 - Prob. 98ESCh. 3.1 - Prob. 99ESCh. 3.1 - Prob. 100ESCh. 3.1 - Prob. 101ESCh. 3.1 - Prob. 102ESCh. 3.1 - Prob. 103ESCh. 3.1 - Prob. 104ESCh. 3.1 - Prob. 105ESCh. 3.1 - Prob. 106ESCh. 3.1 - Prob. 107ESCh. 3.1 - Prob. 108ESCh. 3.1 - Prob. 109ESCh. 3.1 - Prob. 110ESCh. 3.1 - In Exercises 99-132, write an equation to...Ch. 3.1 - Prob. 112ESCh. 3.1 - Prob. 113ESCh. 3.1 - Prob. 114ESCh. 3.1 - Prob. 115ESCh. 3.1 - Prob. 116ESCh. 3.1 - Prob. 117ESCh. 3.1 - Prob. 118ESCh. 3.1 - Prob. 119ESCh. 3.1 - Prob. 120ESCh. 3.1 - Prob. 121ESCh. 3.1 - Prob. 122ESCh. 3.1 - Prob. 123ESCh. 3.1 - Prob. 124ESCh. 3.1 - Prob. 125ESCh. 3.1 - Prob. 126ESCh. 3.1 - Prob. 127ESCh. 3.1 - Prob. 128ESCh. 3.1 - Prob. 129ESCh. 3.1 - Prob. 130ESCh. 3.1 - Prob. 131ESCh. 3.1 - Prob. 132ESCh. 3.1 - Time a. Write an algebraic expression for the...Ch. 3.1 - Prob. 134ESCh. 3.1 - Prob. 135ESCh. 3.1 - Prob. 136ESCh. 3.1 - Prob. 137ESCh. 3.1 - Prob. 138ESCh. 3.1 - Prob. 139ESCh. 3.1 - Solve the inequality 2x43. Graph the solution on a...Ch. 3.2 - Prob. 1ESCh. 3.2 - Prob. 2ESCh. 3.2 - Prob. 3ESCh. 3.2 - Prob. 4ESCh. 3.2 - Prob. 5ESCh. 3.2 - Prob. 6ESCh. 3.2 - Prob. 7ESCh. 3.2 - Prob. 8ESCh. 3.2 - Prob. 9ESCh. 3.2 - Prob. 10ESCh. 3.2 - Prob. 11ESCh. 3.2 - Prob. 12ESCh. 3.2 - Prob. 13ESCh. 3.2 - Prob. 14ESCh. 3.2 - Prob. 15ESCh. 3.2 - Prob. 16ESCh. 3.2 - Prob. 17ESCh. 3.2 - Prob. 18ESCh. 3.2 - Prob. 19ESCh. 3.2 - Prob. 20ESCh. 3.2 - Prob. 21ESCh. 3.2 - Prob. 22ESCh. 3.2 - Prob. 23ESCh. 3.2 - Exercises 7-32 involve determining a number or...Ch. 3.2 - Prob. 25ESCh. 3.2 - Prob. 26ESCh. 3.2 - Prob. 27ESCh. 3.2 - Prob. 28ESCh. 3.2 - Exercises 7-32 involve determining a number or...Ch. 3.2 - Prob. 30ESCh. 3.2 - Prob. 31ESCh. 3.2 - Prob. 32ESCh. 3.2 - Prob. 33ESCh. 3.2 - Prob. 34ESCh. 3.2 - Prob. 35ESCh. 3.2 - Prob. 36ESCh. 3.2 - Prob. 37ESCh. 3.2 - Prob. 38ESCh. 3.2 - Prob. 39ESCh. 3.2 - Prob. 40ESCh. 3.2 - Prob. 41ESCh. 3.2 - Prob. 42ESCh. 3.2 - Prob. 43ESCh. 3.2 - Prob. 44ESCh. 3.2 - Prob. 45ESCh. 3.2 - Prob. 46ESCh. 3.2 - Prob. 47ESCh. 3.2 - Prob. 48ESCh. 3.2 - Prob. 49ESCh. 3.2 - Prob. 50ESCh. 3.2 - Prob. 51ESCh. 3.2 - Prob. 52ESCh. 3.2 - Prob. 53ESCh. 3.2 - Prob. 54ESCh. 3.2 - Prob. 55ESCh. 3.2 - Prob. 56ESCh. 3.2 - Prob. 57ESCh. 3.2 - Prob. 58ESCh. 3.2 - Prob. 59ESCh. 3.2 - Prob. 60ESCh. 3.2 - Exercises 47-68 involve percents. Read Examples...Ch. 3.2 - Prob. 62ESCh. 3.2 - Prob. 63ESCh. 3.2 - Prob. 64ESCh. 3.2 - Prob. 65ESCh. 3.2 - Prob. 66ESCh. 3.2 - Prob. 67ESCh. 3.2 - Prob. 68ESCh. 3.2 - Prob. 69ESCh. 3.2 - Prob. 70ESCh. 3.2 - Prob. 71ESCh. 3.2 - Prob. 72ESCh. 3.2 - Prob. 73ESCh. 3.2 - Prob. 74ESCh. 3.2 - Prob. 75ESCh. 3.2 - Prob. 76ESCh. 3.3 - Prob. 1ESCh. 3.3 - Prob. 2ESCh. 3.3 - Prob. 3ESCh. 3.3 - Prob. 4ESCh. 3.3 - Prob. 5ESCh. 3.3 - Prob. 6ESCh. 3.3 - Prob. 7ESCh. 3.3 - Fill the blanks with the appropriate word, phrase,...Ch. 3.3 - Prob. 9ESCh. 3.3 - Prob. 10ESCh. 3.3 - Triangular Building This building in New York...Ch. 3.3 - Prob. 12ESCh. 3.3 - Prob. 13ESCh. 3.3 - Prob. 14ESCh. 3.3 - Prob. 15ESCh. 3.3 - Prob. 16ESCh. 3.3 - Prob. 17ESCh. 3.3 - Prob. 18ESCh. 3.3 - Prob. 19ESCh. 3.3 - Prob. 20ESCh. 3.3 - Prob. 21ESCh. 3.3 - Prob. 22ESCh. 3.3 - Prob. 23ESCh. 3.3 - Prob. 24ESCh. 3.3 - Prob. 25ESCh. 3.3 - Prob. 26ESCh. 3.3 - Prob. 27ESCh. 3.3 - Parallelogram The two smaller angles of a...Ch. 3.3 - Prob. 29ESCh. 3.3 - Prob. 30ESCh. 3.3 - Prob. 31ESCh. 3.3 - Prob. 32ESCh. 3.3 - Building a Bookcase A bookcase is have four...Ch. 3.3 - Prob. 34ESCh. 3.3 - Prob. 35ESCh. 3.3 - Prob. 36ESCh. 3.3 - Prob. 37ESCh. 3.3 - Storage Shelves Carlotta plans to build storage...Ch. 3.3 - Prob. 39ESCh. 3.3 - Prob. 40ESCh. 3.3 - Prob. 41ESCh. 3.3 - Prob. 42ESCh. 3.3 - Prob. 43ESCh. 3.3 - Prob. 44ESCh. 3.3 - Simplify 7y+x3(x2)+2y.Ch. 3.3 - Prob. 46ESCh. 3.4 - Fill in the blanks with appropriate word, phrase,...Ch. 3.4 - Fill in the blanks with appropriate word, phrase,...Ch. 3.4 - Prob. 3ESCh. 3.4 - Prob. 4ESCh. 3.4 - Prob. 5ESCh. 3.4 - Prob. 6ESCh. 3.4 - Prob. 7ESCh. 3.4 - Prob. 8ESCh. 3.4 - Prob. 9ESCh. 3.4 - Prob. 10ESCh. 3.4 - In Exercises 7-32, set up an equation that can be...Ch. 3.4 - Prob. 12ESCh. 3.4 - Prob. 13ESCh. 3.4 - Prob. 14ESCh. 3.4 - Prob. 15ESCh. 3.4 - Prob. 16ESCh. 3.4 - Prob. 17ESCh. 3.4 - Prob. 18ESCh. 3.4 - Prob. 19ESCh. 3.4 - Prob. 20ESCh. 3.4 - Prob. 21ESCh. 3.4 - Prob. 22ESCh. 3.4 - Prob. 23ESCh. 3.4 - Prob. 24ESCh. 3.4 - Prob. 25ESCh. 3.4 - Prob. 26ESCh. 3.4 - Prob. 27ESCh. 3.4 - Prob. 28ESCh. 3.4 - Prob. 29ESCh. 3.4 - Prob. 30ESCh. 3.4 - Prob. 31ESCh. 3.4 - Prob. 32ESCh. 3.4 - Prob. 33ESCh. 3.4 - Prob. 34ESCh. 3.4 - Prob. 35ESCh. 3.4 - Prob. 36ESCh. 3.4 - Prob. 37ESCh. 3.4 - Prob. 38ESCh. 3.4 - Prob. 39ESCh. 3.4 - Prob. 40ESCh. 3.4 - Prob. 41ESCh. 3.4 - Prob. 42ESCh. 3.4 - Prob. 43ESCh. 3.4 - In Exercises 33-46, set up an equation that can be...Ch. 3.4 - Prob. 45ESCh. 3.4 - Prob. 46ESCh. 3.4 - Prob. 47ESCh. 3.4 - Prob. 48ESCh. 3.4 - Prob. 49ESCh. 3.4 - Prob. 50ESCh. 3.4 - Prob. 51ESCh. 3.4 - Prob. 52ESCh. 3.4 - Prob. 53ESCh. 3.4 - Prob. 54ESCh. 3.4 - Prob. 55ESCh. 3.4 - Prob. 56ESCh. 3.4 - In Exercises 47-66, set up an equation that can be...Ch. 3.4 - Prob. 58ESCh. 3.4 - Prob. 59ESCh. 3.4 - Prob. 60ESCh. 3.4 - Prob. 61ESCh. 3.4 - Prob. 62ESCh. 3.4 - Prob. 63ESCh. 3.4 - Prob. 64ESCh. 3.4 - Prob. 65ESCh. 3.4 - Prob. 66ESCh. 3.4 - Prob. 67ESCh. 3.4 - Prob. 68ESCh. 3.4 - Discuss and answer Exercises 68 and 69 as a group....Ch. 3.4 - Prob. 70ESCh. 3.4 - Prob. 71ESCh. 3.4 - Prob. 72ESCh. 3.4 - Prob. 73ESCh. 3 - In Exercises 1-6, express each statement as an...Ch. 3 - Prob. 2MCTCh. 3 - Prob. 3MCTCh. 3 - Prob. 4MCTCh. 3 - In Exercises 1-6, express each statement as an...Ch. 3 - In Exercises 1-6, express each statement as an...Ch. 3 - Prob. 7MCTCh. 3 - Prob. 8MCTCh. 3 - Select a variable to represent one quantity and...Ch. 3 - Prob. 10MCTCh. 3 - In Exercises 10-12, write the problem as an...Ch. 3 - Prob. 12MCTCh. 3 - Prob. 13MCTCh. 3 - Prob. 14MCTCh. 3 - Prob. 15MCTCh. 3 - Prob. 16MCTCh. 3 - Prob. 17MCTCh. 3 - Prob. 18MCTCh. 3 - Prob. 19MCTCh. 3 - Prob. 20MCTCh. 3 - Prob. 1RECh. 3 - Prob. 2RECh. 3 - Prob. 3RECh. 3 - Prob. 4RECh. 3 - Prob. 5RECh. 3 - Prob. 6RECh. 3 - Prob. 7RECh. 3 - Prob. 8RECh. 3 - Prob. 9RECh. 3 - Prob. 10RECh. 3 - Prob. 11RECh. 3 - Prob. 12RECh. 3 - Prob. 13RECh. 3 - In Exercises 9-18, set up an equation that can be...Ch. 3 - Prob. 15RECh. 3 - Prob. 16RECh. 3 - Prob. 17RECh. 3 - Prob. 18RECh. 3 - Prob. 19RECh. 3 - Prob. 20RECh. 3 - Prob. 21RECh. 3 - Designing a House The figure at right, above shows...Ch. 3 - Rhombus The two larger angles of a rhombus are...Ch. 3 - Bookcase Wade is building a bookcase with 4...Ch. 3 - Prob. 25RECh. 3 - Trains Leaving Two trains going in opposite...Ch. 3 - Prob. 27RECh. 3 - Prob. 28RECh. 3 - Prob. 29RECh. 3 - Prob. 30RECh. 3 - Prob. 31RECh. 3 - Prob. 32RECh. 3 - Prob. 33RECh. 3 - Prob. 34RECh. 3 - Prob. 35RECh. 3 - Prob. 36RECh. 3 - Prob. 37RECh. 3 - Prob. 38RECh. 3 - Prob. 39RECh. 3 - Prob. 40RECh. 3 - Prob. 41RECh. 3 - Speed Traveled Two brothers who are 230 miles...Ch. 3 - Prob. 43RECh. 3 - Prob. 44RECh. 3 - Prob. 45RECh. 3 - Prob. 1PTCh. 3 - Prob. 2PTCh. 3 - Prob. 3PTCh. 3 - Prob. 4PTCh. 3 - Prob. 5PTCh. 3 - Prob. 6PTCh. 3 - Prob. 7PTCh. 3 - Prob. 8PTCh. 3 - Prob. 9PTCh. 3 - Prob. 10PTCh. 3 - Prob. 11PTCh. 3 - Prob. 12PTCh. 3 - Prob. 13PTCh. 3 - Prob. 14PTCh. 3 - Prob. 15PTCh. 3 - Prob. 16PTCh. 3 - Prob. 17PTCh. 3 - Prob. 18PTCh. 3 - Prob. 19PTCh. 3 - Prob. 20PTCh. 3 - Prob. 21PTCh. 3 - Prob. 22PTCh. 3 - Prob. 23PTCh. 3 - Prob. 24PTCh. 3 - Prob. 25PTCh. 3 - Prob. 1CRTCh. 3 - Prob. 2CRTCh. 3 - Prob. 3CRTCh. 3 - Prob. 4CRTCh. 3 - Prob. 5CRTCh. 3 - Prob. 6CRTCh. 3 - Prob. 7CRTCh. 3 - Prob. 8CRTCh. 3 - Prob. 9CRTCh. 3 - Prob. 10CRTCh. 3 - Prob. 11CRTCh. 3 - Prob. 12CRTCh. 3 - Prob. 13CRTCh. 3 - Prob. 14CRTCh. 3 - Prob. 15CRTCh. 3 - Prob. 16CRTCh. 3 - Prob. 17CRTCh. 3 - Prob. 18CRTCh. 3 - Prob. 19CRTCh. 3 - Quadrilateral One angle of a quadrilateral...
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