College Algebra (10th Edition)
10th Edition
ISBN: 9780321979476
Author: Michael Sullivan
Publisher: PEARSON
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Question
Chapter 3.3, Problem 38SB
To determine
To find: The given function whether even, odd or neither.
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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
College Algebra (10th Edition)
Ch. 3.1 - Prob. 1AYPCh. 3.1 - Prob. 2AYPCh. 3.1 - Prob. 3AYPCh. 3.1 - Prob. 4AYPCh. 3.1 - Prob. 5AYPCh. 3.1 - Prob. 6AYPCh. 3.1 - Prob. 7CVCh. 3.1 - Prob. 8CVCh. 3.1 - Prob. 9CVCh. 3.1 - Prob. 10CV
Ch. 3.1 - Prob. 11CVCh. 3.1 - Prob. 12CVCh. 3.1 - Prob. 13CVCh. 3.1 - Prob. 14CVCh. 3.1 - Prob. 15CVCh. 3.1 - Prob. 16CVCh. 3.1 - Prob. 17CVCh. 3.1 - Prob. 18CVCh. 3.1 - Prob. 19SBCh. 3.1 - Prob. 20SBCh. 3.1 - Prob. 21SBCh. 3.1 - Prob. 22SBCh. 3.1 - Prob. 23SBCh. 3.1 - Prob. 24SBCh. 3.1 - Prob. 25SBCh. 3.1 - Prob. 26SBCh. 3.1 - Prob. 27SBCh. 3.1 - Prob. 28SBCh. 3.1 - Prob. 29SBCh. 3.1 - Prob. 30SBCh. 3.1 - Prob. 31SBCh. 3.1 - Prob. 32SBCh. 3.1 - Prob. 33SBCh. 3.1 - Prob. 34SBCh. 3.1 - Prob. 35SBCh. 3.1 - Prob. 36SBCh. 3.1 - Prob. 37SBCh. 3.1 - Prob. 38SBCh. 3.1 - Prob. 39SBCh. 3.1 - Prob. 40SBCh. 3.1 - Prob. 41SBCh. 3.1 - Prob. 42SBCh. 3.1 - Prob. 43SBCh. 3.1 - Prob. 44SBCh. 3.1 - Prob. 45SBCh. 3.1 - Prob. 46SBCh. 3.1 - Prob. 47SBCh. 3.1 - Prob. 48SBCh. 3.1 - Prob. 49SBCh. 3.1 - Prob. 50SBCh. 3.1 - Prob. 51SBCh. 3.1 - Prob. 52SBCh. 3.1 - Prob. 53SBCh. 3.1 - Prob. 54SBCh. 3.1 - Prob. 55SBCh. 3.1 - Prob. 56SBCh. 3.1 - Prob. 57SBCh. 3.1 - Prob. 58SBCh. 3.1 - Prob. 59SBCh. 3.1 - Prob. 60SBCh. 3.1 - 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Prob. 111AECh. 3.1 - Prob. 112AECh. 3.1 - Prob. 113AECh. 3.1 - Prob. 114AECh. 3.1 - Prob. 115DWCh. 3.1 - Prob. 116DWCh. 3.1 - Prob. 117DWCh. 3.1 - Prob. 118RYKCh. 3.1 - Prob. 119RYKCh. 3.1 - Prob. 120RYKCh. 3.1 - Prob. 121RYKCh. 3.2 - Prob. 1AYPCh. 3.2 - Prob. 2AYPCh. 3.2 - Prob. 3CVCh. 3.2 - Prob. 4CVCh. 3.2 - Prob. 5CVCh. 3.2 - Prob. 6CVCh. 3.2 - Prob. 7CVCh. 3.2 - Prob. 8CVCh. 3.2 - Prob. 9CVCh. 3.2 - Prob. 10CVCh. 3.2 - Prob. 11SBCh. 3.2 - Prob. 12SBCh. 3.2 - Prob. 13SBCh. 3.2 - Prob. 14SBCh. 3.2 - Prob. 15SBCh. 3.2 - Prob. 16SBCh. 3.2 - Prob. 17SBCh. 3.2 - Prob. 18SBCh. 3.2 - Prob. 19SBCh. 3.2 - Prob. 20SBCh. 3.2 - Prob. 21SBCh. 3.2 - Prob. 22SBCh. 3.2 - Prob. 23SBCh. 3.2 - Prob. 24SBCh. 3.2 - Prob. 25SBCh. 3.2 - Prob. 26SBCh. 3.2 - Prob. 27SBCh. 3.2 - Prob. 28SBCh. 3.2 - Prob. 29SBCh. 3.2 - Prob. 30SBCh. 3.2 - Prob. 31AECh. 3.2 - Prob. 32AECh. 3.2 - Prob. 33AECh. 3.2 - Prob. 34AECh. 3.2 - Prob. 35AECh. 3.2 - Prob. 36AECh. 3.2 - Prob. 37AECh. 3.2 - Prob. 38AECh. 3.2 - Prob. 39AECh. 3.2 - Prob. 40DWCh. 3.2 - Prob. 41DWCh. 3.2 - Prob. 42DWCh. 3.2 - Prob. 43DWCh. 3.2 - Prob. 44DWCh. 3.2 - Prob. 45DWCh. 3.2 - Prob. 46DWCh. 3.2 - Prob. 47DWCh. 3.2 - Prob. 48DWCh. 3.2 - Prob. 49DWCh. 3.2 - Prob. 50DWCh. 3.2 - Prob. 51DWCh. 3.2 - Prob. 52RYKCh. 3.2 - Prob. 53RYKCh. 3.2 - Prob. 54RYKCh. 3.2 - Prob. 55RYKCh. 3.3 - Prob. 1AYPCh. 3.3 - Prob. 2AYPCh. 3.3 - Prob. 3AYPCh. 3.3 - Prob. 4AYPCh. 3.3 - Prob. 5AYPCh. 3.3 - Prob. 6CVCh. 3.3 - Prob. 7CVCh. 3.3 - Prob. 8CVCh. 3.3 - Prob. 9CVCh. 3.3 - Prob. 10CVCh. 3.3 - Prob. 11CVCh. 3.3 - Prob. 12CVCh. 3.3 - Prob. 13SBCh. 3.3 - Prob. 14SBCh. 3.3 - Prob. 15SBCh. 3.3 - Prob. 16SBCh. 3.3 - Prob. 17SBCh. 3.3 - Prob. 18SBCh. 3.3 - Prob. 19SBCh. 3.3 - Prob. 20SBCh. 3.3 - Prob. 21SBCh. 3.3 - Prob. 22SBCh. 3.3 - Prob. 23SBCh. 3.3 - Prob. 24SBCh. 3.3 - Prob. 25SBCh. 3.3 - Prob. 26SBCh. 3.3 - Prob. 27SBCh. 3.3 - Prob. 28SBCh. 3.3 - Prob. 29SBCh. 3.3 - Prob. 30SBCh. 3.3 - Prob. 31SBCh. 3.3 - Prob. 32SBCh. 3.3 - Prob. 33SBCh. 3.3 - Prob. 34SBCh. 3.3 - 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Prob. 85AECh. 3.3 - Prob. 86AECh. 3.3 - Prob. 87AECh. 3.3 - Prob. 88AECh. 3.3 - Prob. 89AECh. 3.3 - Prob. 90AECh. 3.3 - Prob. 91AECh. 3.3 - Prob. 92AECh. 3.3 - Prob. 93AECh. 3.3 - Prob. 94AECh. 3.3 - Prob. 95DWCh. 3.3 - Prob. 96DWCh. 3.3 - Prob. 97DWCh. 3.3 - Prob. 98DWCh. 3.3 - Prob. 99DWCh. 3.3 - Prob. 100DWCh. 3.3 - Prob. 101DWCh. 3.3 - Prob. 102DWCh. 3.3 - Prob. 103RYKCh. 3.3 - Prob. 104RYKCh. 3.3 - Prob. 105RYKCh. 3.3 - Prob. 106RYKCh. 3.4 - Prob. 1AYPCh. 3.4 - Prob. 2AYPCh. 3.4 - Prob. 3AYPCh. 3.4 - Prob. 4CVCh. 3.4 - Prob. 5CVCh. 3.4 - Prob. 6CVCh. 3.4 - Prob. 7CVCh. 3.4 - Prob. 8CVCh. 3.4 - Prob. 9CVCh. 3.4 - Prob. 10CVCh. 3.4 - Prob. 11SBCh. 3.4 - Prob. 12SBCh. 3.4 - Prob. 13SBCh. 3.4 - Prob. 14SBCh. 3.4 - Prob. 15SBCh. 3.4 - Prob. 16SBCh. 3.4 - Prob. 17SBCh. 3.4 - Prob. 18SBCh. 3.4 - Prob. 19SBCh. 3.4 - Prob. 20SBCh. 3.4 - Prob. 21SBCh. 3.4 - Prob. 22SBCh. 3.4 - Prob. 23SBCh. 3.4 - Prob. 24SBCh. 3.4 - Prob. 25SBCh. 3.4 - Prob. 26SBCh. 3.4 - Prob. 27SBCh. 3.4 - Prob. 28SBCh. 3.4 - Prob. 29SBCh. 3.4 - Prob. 30SBCh. 3.4 - Prob. 31SBCh. 3.4 - Prob. 32SBCh. 3.4 - Prob. 33SBCh. 3.4 - Prob. 34SBCh. 3.4 - Prob. 35SBCh. 3.4 - Prob. 36SBCh. 3.4 - Prob. 37SBCh. 3.4 - Prob. 38SBCh. 3.4 - Prob. 39SBCh. 3.4 - Prob. 40SBCh. 3.4 - Prob. 41SBCh. 3.4 - Prob. 42SBCh. 3.4 - Prob. 43SBCh. 3.4 - Prob. 44SBCh. 3.4 - Prob. 45SBCh. 3.4 - Prob. 46SBCh. 3.4 - Prob. 47SBCh. 3.4 - Prob. 48SBCh. 3.4 - Prob. 49AECh. 3.4 - Prob. 50AECh. 3.4 - Prob. 51AECh. 3.4 - Prob. 52AECh. 3.4 - Prob. 53AECh. 3.4 - Prob. 54AECh. 3.4 - Prob. 55AECh. 3.4 - Prob. 56AECh. 3.4 - Prob. 57AECh. 3.4 - Prob. 58AECh. 3.4 - Prob. 59AECh. 3.4 - Prob. 60AECh. 3.4 - Prob. 61AECh. 3.4 - Prob. 62DWCh. 3.4 - Prob. 63DWCh. 3.4 - Prob. 64DWCh. 3.4 - Prob. 65DWCh. 3.4 - Prob. 66DWCh. 3.4 - Prob. 67DWCh. 3.4 - Prob. 68DWCh. 3.4 - Prob. 69DWCh. 3.4 - Prob. 70DWCh. 3.4 - Prob. 71DWCh. 3.4 - Prob. 72RYKCh. 3.4 - Prob. 73RYKCh. 3.4 - Prob. 74RYKCh. 3.4 - Prob. 75RYKCh. 3.5 - Prob. 1CVCh. 3.5 - Prob. 2CVCh. 3.5 - Prob. 3CVCh. 3.5 - Prob. 4CVCh. 3.5 - Prob. 5CVCh. 3.5 - Prob. 6CVCh. 3.5 - Prob. 7SBCh. 3.5 - Prob. 8SBCh. 3.5 - Prob. 9SBCh. 3.5 - Prob. 10SBCh. 3.5 - Prob. 11SBCh. 3.5 - Prob. 12SBCh. 3.5 - Prob. 13SBCh. 3.5 - Prob. 14SBCh. 3.5 - Prob. 15SBCh. 3.5 - Prob. 16SBCh. 3.5 - Prob. 17SBCh. 3.5 - Prob. 18SBCh. 3.5 - Prob. 19SBCh. 3.5 - Prob. 20SBCh. 3.5 - Prob. 21SBCh. 3.5 - Prob. 22SBCh. 3.5 - Prob. 23SBCh. 3.5 - Prob. 24SBCh. 3.5 - Prob. 25SBCh. 3.5 - Prob. 26SBCh. 3.5 - Prob. 27SBCh. 3.5 - Prob. 28SBCh. 3.5 - Prob. 29SBCh. 3.5 - Prob. 30SBCh. 3.5 - Prob. 31SBCh. 3.5 - Prob. 32SBCh. 3.5 - Prob. 33SBCh. 3.5 - Prob. 34SBCh. 3.5 - Prob. 35SBCh. 3.5 - Prob. 36SBCh. 3.5 - Prob. 37SBCh. 3.5 - Prob. 38SBCh. 3.5 - Prob. 39SBCh. 3.5 - Prob. 40SBCh. 3.5 - Prob. 41SBCh. 3.5 - Prob. 42SBCh. 3.5 - Prob. 43SBCh. 3.5 - Prob. 44SBCh. 3.5 - Prob. 45SBCh. 3.5 - Prob. 46SBCh. 3.5 - Prob. 47SBCh. 3.5 - Prob. 48SBCh. 3.5 - Prob. 49SBCh. 3.5 - Prob. 50SBCh. 3.5 - Prob. 51SBCh. 3.5 - Prob. 52SBCh. 3.5 - Prob. 53SBCh. 3.5 - Prob. 54SBCh. 3.5 - Prob. 55SBCh. 3.5 - Prob. 56SBCh. 3.5 - Prob. 57SBCh. 3.5 - Prob. 58SBCh. 3.5 - Prob. 59SBCh. 3.5 - Prob. 60SBCh. 3.5 - Prob. 61SBCh. 3.5 - Prob. 62SBCh. 3.5 - Prob. 63SBCh. 3.5 - Prob. 64SBCh. 3.5 - Prob. 65SBCh. 3.5 - Prob. 66SBCh. 3.5 - Prob. 67MPCh. 3.5 - Prob. 68MPCh. 3.5 - Prob. 69MPCh. 3.5 - Prob. 70MPCh. 3.5 - Prob. 71MPCh. 3.5 - Prob. 72MPCh. 3.5 - Prob. 73MPCh. 3.5 - Prob. 74MPCh. 3.5 - Prob. 75AECh. 3.5 - Prob. 76AECh. 3.5 - Prob. 77AECh. 3.5 - Prob. 78AECh. 3.5 - Prob. 79AECh. 3.5 - Prob. 80AECh. 3.5 - Prob. 81AECh. 3.5 - Prob. 82AECh. 3.5 - Prob. 83AECh. 3.5 - Prob. 84AECh. 3.5 - Prob. 85AECh. 3.5 - Prob. 86AECh. 3.5 - Prob. 87AECh. 3.5 - Prob. 88AECh. 3.5 - Prob. 89DWCh. 3.5 - Prob. 90DWCh. 3.5 - Prob. 91DWCh. 3.5 - Prob. 92DWCh. 3.5 - Prob. 93DWCh. 3.5 - Prob. 94RYKCh. 3.5 - Prob. 95RYKCh. 3.5 - Prob. 96RYKCh. 3.5 - Prob. 97RYKCh. 3.6 - Prob. 1AECh. 3.6 - Prob. 2AECh. 3.6 - Prob. 3AECh. 3.6 - Prob. 4AECh. 3.6 - Prob. 5AECh. 3.6 - Prob. 6AECh. 3.6 - Prob. 7AECh. 3.6 - Prob. 8AECh. 3.6 - 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