ALEKS 360 Access Card (18 weeks) for Intermediate Algebra
5th Edition
ISBN: 9781259948794
Author: Julie Miller, Molly O'Neill, Nancy Hyde
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Textbook Question
Chapter R4, Problem 6PE
For Exercises 5-8, write the set in interval notation.
6.
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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 R4 Solutions
ALEKS 360 Access Card (18 weeks) for Intermediate Algebra
Ch. R4 - Prob. 1PECh. R4 - Prob. 2PECh. R4 - Prob. 3PECh. R4 - For Exercises 2-4, simplify.
4.
Ch. R4 - Prob. 5PECh. R4 - For Exercises 5-8, write the set in interval...Ch. R4 - Prob. 7PECh. R4 - Prob. 8PECh. R4 - For Exercises 9-12:
a. Determine the number of...Ch. R4 - Prob. 10PE
Ch. R4 - Prob. 11PECh. R4 - Prob. 12PECh. R4 - Prob. 13PECh. R4 - Prob. 14PECh. R4 - Prob. 15PECh. R4 - Prob. 16PECh. R4 - Prob. 17PECh. R4 - Prob. 18PECh. R4 - Prob. 19PECh. R4 - Prob. 20PECh. R4 - Prob. 21PECh. R4 - Prob. 22PECh. R4 - Prob. 23PECh. R4 - Prob. 24PECh. R4 - Prob. 25PECh. R4 - Prob. 26PECh. R4 - Prob. 27PECh. R4 - Prob. 28PECh. R4 - Prob. 29PECh. R4 - Prob. 30PECh. R4 - Prob. 31PECh. R4 - Prob. 32PECh. R4 - Prob. 33PECh. R4 - Prob. 34PECh. R4 - Prob. 35PECh. R4 - Prob. 36PECh. R4 - Prob. 37PECh. R4 - Prob. 38PECh. R4 - Prob. 39PECh. R4 - Prob. 40PECh. R4 - Prob. 41PECh. R4 - Prob. 42PECh. R4 - Prob. 43PECh. R4 - Prob. 44PECh. R4 - Prob. 45PECh. R4 - Prob. 46PECh. R4 - Prob. 47PECh. R4 - Prob. 48PECh. R4 - Prob. 49PECh. R4 - Prob. 50PECh. R4 - Prob. 51PECh. R4 - Prob. 52PECh. R4 - Prob. 53PECh. R4 - Prob. 54PECh. R4 - Prob. 55PECh. R4 - Prob. 56PECh. R4 - Prob. 57PECh. R4 - Prob. 58PECh. R4 - Prob. 59PECh. R4 - Prob. 60PECh. R4 - Prob. 61PECh. R4 - Prob. 62PECh. R4 - Prob. 63PECh. R4 - Prob. 64PECh. R4 - Prob. 65PECh. R4 - Prob. 66PECh. R4 - Prob. 67PECh. R4 - Prob. 68PECh. R4 - Prob. 69PECh. R4 - Prob. 70PECh. R4 - Prob. 71PECh. R4 - Prob. 72PECh. R4 - Prob. 73PECh. R4 - Prob. 74PECh. R4 - Prob. 75PECh. R4 - Prob. 76PECh. R4 - Prob. 77PECh. R4 - Prob. 78PECh. R4 - Prob. 79PECh. R4 - Prob. 80PECh. R4 - Prob. 81PECh. R4 - Prob. 82PECh. R4 - Prob. 83PECh. R4 - Prob. 84PECh. R4 - Prob. 85PECh. R4 - Prob. 86PECh. R4 - Prob. 87PE
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