Pearson eText Algebra and Trigonometry -- Instant Access (Pearson+)
11th Edition
ISBN: 9780136872689
Author: Michael Sullivan
Publisher: PEARSON+
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Question
Chapter 7.1, Problem 61AYU
To determine
To convert: The angle
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Students have asked these similar questions
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]
Question 4
(a) The following matrices represent linear maps on R² with respect to an
orthonormal basis:
=
[1/√5 2/√5
[2/√5 -1/√5]
"
[1/√5 2/√5]
A =
B =
[2/√5 1/√5] 1
C =
D =
=
=
[ 1/3/5 2/35]
1/√5 2/√5
-2/√5 1/√5'
For each of the matrices A, B, C, D, state whether it represents a self-adjoint
linear map, an orthogonal linear map, both, or neither.
(b) For the quadratic form
q(x, y, z) = y² + 2xy +2yz
over R, write down a linear change of variables to u, v, w such that q in these
terms is in canonical form for Sylvester's Law of Inertia.
[6]
[4]
Chapter 7 Solutions
Pearson eText Algebra and Trigonometry -- Instant Access (Pearson+)
Ch. 7.1 - Prob. 1AYUCh. 7.1 - Prob. 2AYUCh. 7.1 - Prob. 3AYUCh. 7.1 - Prob. 4AYUCh. 7.1 - Prob. 5AYUCh. 7.1 - Prob. 6AYUCh. 7.1 - Prob. 7AYUCh. 7.1 - Prob. 8AYUCh. 7.1 - Prob. 9AYUCh. 7.1 - Prob. 10AYU
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Prob. 102AYUCh. 7.3 - Prob. 103AYUCh. 7.3 - Prob. 104AYUCh. 7.3 - Prob. 105AYUCh. 7.3 - Prob. 106AYUCh. 7.4 - Prob. 1AYUCh. 7.4 - Prob. 2AYUCh. 7.4 - Prob. 3AYUCh. 7.4 - Prob. 4AYUCh. 7.4 - Prob. 5AYUCh. 7.4 - Prob. 6AYUCh. 7.4 - Prob. 7AYUCh. 7.4 - Prob. 8AYUCh. 7.4 - Prob. 9AYUCh. 7.4 - Prob. 10AYUCh. 7.4 - Prob. 11AYUCh. 7.4 - Prob. 12AYUCh. 7.4 - Prob. 13AYUCh. 7.4 - Prob. 14AYUCh. 7.4 - Prob. 15AYUCh. 7.4 - Prob. 16AYUCh. 7.4 - Prob. 17AYUCh. 7.4 - Prob. 18AYUCh. 7.4 - Prob. 19AYUCh. 7.4 - Prob. 20AYUCh. 7.4 - Prob. 21AYUCh. 7.4 - Prob. 22AYUCh. 7.4 - Prob. 23AYUCh. 7.4 - Prob. 24AYUCh. 7.4 - Prob. 25AYUCh. 7.4 - Prob. 26AYUCh. 7.4 - Prob. 27AYUCh. 7.4 - Prob. 28AYUCh. 7.4 - Prob. 29AYUCh. 7.4 - Prob. 30AYUCh. 7.4 - Prob. 31AYUCh. 7.4 - Prob. 32AYUCh. 7.4 - Prob. 33AYUCh. 7.4 - Prob. 34AYUCh. 7.4 - Prob. 35AYUCh. 7.4 - Prob. 36AYUCh. 7.4 - Prob. 37AYUCh. 7.4 - Prob. 38AYUCh. 7.4 - Prob. 39AYUCh. 7.4 - Prob. 40AYUCh. 7.4 - Prob. 41AYUCh. 7.4 - Prob. 42AYUCh. 7.4 - Prob. 43AYUCh. 7.4 - 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Prob. 53AYUCh. 7.5 - Prob. 54AYUCh. 7.5 - Prob. 55AYUCh. 7.5 - Prob. 56AYUCh. 7.5 - Prob. 57AYUCh. 7.5 - Prob. 58AYUCh. 7.5 - Prob. 59AYUCh. 7.5 - Prob. 60AYUCh. 7.5 - Prob. 61AYUCh. 7.5 - Prob. 62AYUCh. 7.5 - Prob. 63AYUCh. 7.5 - Prob. 64AYUCh. 7.5 - Prob. 65AYUCh. 7.5 - Prob. 66AYUCh. 7.5 - Prob. 67AYUCh. 7.5 - Prob. 68AYUCh. 7.5 - Prob. 69AYUCh. 7.5 - Prob. 70AYUCh. 7.5 - Prob. 71AYUCh. 7.5 - Prob. 72AYUCh. 7.5 - Prob. 73AYUCh. 7.5 - Prob. 74AYUCh. 7.5 - Prob. 75AYUCh. 7.5 - Prob. 76AYUCh. 7.5 - Prob. 77AYUCh. 7.5 - Prob. 78AYUCh. 7.5 - Prob. 79AYUCh. 7.5 - Prob. 80AYUCh. 7.5 - Prob. 81AYUCh. 7.5 - Prob. 82AYUCh. 7.5 - Prob. 83AYUCh. 7.5 - Prob. 84AYUCh. 7.5 - Prob. 85AYUCh. 7.5 - Prob. 86AYUCh. 7.5 - Prob. 87AYUCh. 7.5 - Prob. 88AYUCh. 7.5 - Prob. 89AYUCh. 7.5 - Prob. 90AYUCh. 7.5 - Prob. 91AYUCh. 7.5 - Prob. 92AYUCh. 7.5 - Prob. 93AYUCh. 7.5 - Prob. 94AYUCh. 7.5 - Prob. 95AYUCh. 7.5 - Prob. 96AYUCh. 7.5 - Prob. 97AYUCh. 7.5 - Prob. 98AYUCh. 7.5 - Prob. 99AYUCh. 7.5 - Prob. 100AYUCh. 7.5 - Prob. 101AYUCh. 7.5 - Prob. 102AYUCh. 7.5 - Prob. 103AYUCh. 7.5 - Prob. 104AYUCh. 7.5 - Prob. 105AYUCh. 7.5 - Prob. 106AYUCh. 7.5 - Prob. 107AYUCh. 7.5 - Prob. 108AYUCh. 7.5 - Prob. 109AYUCh. 7.5 - Prob. 110AYUCh. 7.5 - Prob. 111AYUCh. 7.5 - Prob. 112AYUCh. 7.5 - Prob. 113AYUCh. 7.5 - Prob. 114AYUCh. 7.5 - Prob. 115AYUCh. 7.5 - Prob. 116AYUCh. 7.5 - Prob. 117AYUCh. 7.5 - Prob. 118AYUCh. 7.5 - Prob. 119AYUCh. 7.5 - Prob. 120AYUCh. 7.5 - Prob. 121AYUCh. 7.6 - Prob. 1AYUCh. 7.6 - Prob. 2AYUCh. 7.6 - Prob. 3AYUCh. 7.6 - Prob. 4AYUCh. 7.6 - Prob. 5AYUCh. 7.6 - Prob. 6AYUCh. 7.6 - Prob. 7AYUCh. 7.6 - Prob. 8AYUCh. 7.6 - Prob. 9AYUCh. 7.6 - Prob. 10AYUCh. 7.6 - Prob. 11AYUCh. 7.6 - Prob. 12AYUCh. 7.6 - Prob. 13AYUCh. 7.6 - Prob. 14AYUCh. 7.6 - Prob. 15AYUCh. 7.6 - Prob. 16AYUCh. 7.6 - Prob. 17AYUCh. 7.6 - Prob. 18AYUCh. 7.6 - Prob. 19AYUCh. 7.6 - Prob. 20AYUCh. 7.6 - Prob. 21AYUCh. 7.6 - Prob. 22AYUCh. 7.6 - Prob. 23AYUCh. 7.6 - Prob. 24AYUCh. 7.6 - Prob. 25AYUCh. 7.6 - Prob. 26AYUCh. 7.6 - Prob. 27AYUCh. 7.6 - Prob. 28AYUCh. 7.6 - Prob. 29AYUCh. 7.6 - Prob. 30AYUCh. 7.6 - Prob. 31AYUCh. 7.6 - Prob. 32AYUCh. 7.6 - Prob. 33AYUCh. 7.6 - Prob. 34AYUCh. 7.6 - Prob. 35AYUCh. 7.6 - Prob. 36AYUCh. 7.6 - Prob. 37AYUCh. 7.6 - Prob. 38AYUCh. 7.6 - Prob. 39AYUCh. 7.6 - Prob. 40AYUCh. 7.6 - Prob. 41AYUCh. 7.6 - Prob. 42AYUCh. 7.6 - Prob. 43AYUCh. 7.6 - Prob. 44AYUCh. 7.6 - Prob. 45AYUCh. 7.6 - Prob. 46AYUCh. 7.6 - Prob. 47AYUCh. 7.6 - Prob. 48AYUCh. 7.6 - Prob. 49AYUCh. 7.6 - Prob. 50AYUCh. 7.6 - Prob. 51AYUCh. 7.6 - Prob. 52AYUCh. 7.6 - Prob. 53AYUCh. 7.6 - Prob. 54AYUCh. 7.6 - Prob. 55AYUCh. 7.6 - Prob. 56AYUCh. 7.6 - Prob. 57AYUCh. 7.6 - Prob. 58AYUCh. 7.6 - Prob. 59AYUCh. 7.6 - Prob. 60AYUCh. 7.6 - Prob. 61AYUCh. 7.6 - Prob. 62AYUCh. 7.6 - Prob. 63AYUCh. 7.6 - Prob. 64AYUCh. 7.6 - Prob. 65AYUCh. 7.6 - Prob. 66AYUCh. 7.6 - Prob. 67AYUCh. 7.6 - Prob. 68AYUCh. 7.6 - Prob. 69AYUCh. 7.6 - Prob. 70AYUCh. 7.6 - Prob. 71AYUCh. 7.6 - Prob. 72AYUCh. 7.6 - Prob. 73AYUCh. 7.6 - Prob. 74AYUCh. 7.6 - Prob. 75AYUCh. 7.6 - Prob. 76AYUCh. 7.6 - Prob. 77AYUCh. 7.6 - Prob. 78AYUCh. 7.6 - Prob. 79AYUCh. 7.6 - Prob. 80AYUCh. 7.6 - Prob. 81AYUCh. 7.6 - Prob. 82AYUCh. 7.6 - Prob. 83AYUCh. 7.6 - Prob. 84AYUCh. 7.6 - Prob. 85AYUCh. 7.6 - Prob. 86AYUCh. 7.6 - Prob. 87AYUCh. 7.6 - Prob. 88AYUCh. 7.6 - Prob. 89AYUCh. 7.6 - Prob. 90AYUCh. 7.6 - Prob. 91AYUCh. 7.6 - Prob. 92AYUCh. 7.6 - Prob. 93AYUCh. 7.6 - Prob. 94AYUCh. 7.6 - Prob. 95AYUCh. 7.6 - Prob. 96AYUCh. 7.6 - Prob. 97AYUCh. 7.6 - Prob. 98AYUCh. 7.6 - Prob. 99AYUCh. 7.6 - Prob. 100AYUCh. 7.6 - Prob. 101AYUCh. 7.6 - Prob. 102AYUCh. 7.6 - Prob. 103AYUCh. 7.6 - Prob. 104AYUCh. 7.6 - Prob. 105AYUCh. 7.6 - Prob. 106AYUCh. 7.6 - Prob. 107AYUCh. 7.6 - Prob. 108AYUCh. 7.6 - Prob. 109AYUCh. 7.6 - Prob. 110AYUCh. 7.6 - Prob. 111AYUCh. 7.6 - Prob. 112AYUCh. 7.6 - Prob. 113AYUCh. 7.6 - Prob. 114AYUCh. 7.7 - Prob. 1AYUCh. 7.7 - Prob. 2AYUCh. 7.7 - Prob. 3AYUCh. 7.7 - Prob. 4AYUCh. 7.7 - Prob. 5AYUCh. 7.7 - Prob. 6AYUCh. 7.7 - Prob. 7AYUCh. 7.7 - Prob. 8AYUCh. 7.7 - Prob. 9AYUCh. 7.7 - Prob. 10AYUCh. 7.7 - Prob. 11AYUCh. 7.7 - Prob. 12AYUCh. 7.7 - Prob. 13AYUCh. 7.7 - Prob. 14AYUCh. 7.7 - Prob. 15AYUCh. 7.7 - Prob. 16AYUCh. 7.7 - Prob. 17AYUCh. 7.7 - Prob. 18AYUCh. 7.7 - Prob. 19AYUCh. 7.7 - Prob. 20AYUCh. 7.7 - Prob. 21AYUCh. 7.7 - Prob. 22AYUCh. 7.7 - Prob. 23AYUCh. 7.7 - Prob. 24AYUCh. 7.7 - Prob. 25AYUCh. 7.7 - Prob. 26AYUCh. 7.7 - Prob. 27AYUCh. 7.7 - Prob. 28AYUCh. 7.7 - Prob. 29AYUCh. 7.7 - Prob. 30AYUCh. 7.7 - Prob. 31AYUCh. 7.7 - Prob. 32AYUCh. 7.7 - Prob. 33AYUCh. 7.7 - Prob. 34AYUCh. 7.7 - Prob. 35AYUCh. 7.7 - Prob. 36AYUCh. 7.7 - Prob. 37AYUCh. 7.7 - Prob. 38AYUCh. 7.7 - Prob. 39AYUCh. 7.7 - Prob. 40AYUCh. 7.7 - Prob. 41AYUCh. 7.7 - Prob. 42AYUCh. 7.7 - Prob. 43AYUCh. 7.7 - Prob. 44AYUCh. 7.7 - Prob. 45AYUCh. 7.7 - Prob. 46AYUCh. 7.7 - Prob. 47AYUCh. 7.7 - Prob. 48AYUCh. 7.7 - Prob. 49AYUCh. 7.7 - Prob. 50AYUCh. 7.7 - Prob. 51AYUCh. 7.7 - Prob. 52AYUCh. 7.7 - Prob. 53AYUCh. 7.7 - Prob. 54AYUCh. 7.7 - Prob. 55AYUCh. 7.7 - Prob. 56AYUCh. 7.7 - Prob. 57AYUCh. 7.7 - Prob. 58AYUCh. 7.7 - Prob. 59AYUCh. 7.7 - Prob. 60AYUCh. 7.7 - Prob. 61AYUCh. 7.7 - Prob. 62AYUCh. 7.7 - Prob. 63AYUCh. 7.7 - Prob. 64AYUCh. 7.7 - Prob. 65AYUCh. 7.8 - Prob. 1AYUCh. 7.8 - Prob. 2AYUCh. 7.8 - Prob. 3AYUCh. 7.8 - Prob. 4AYUCh. 7.8 - Prob. 5AYUCh. 7.8 - Prob. 6AYUCh. 7.8 - Prob. 7AYUCh. 7.8 - Prob. 8AYUCh. 7.8 - Prob. 9AYUCh. 7.8 - Prob. 10AYUCh. 7.8 - Prob. 11AYUCh. 7.8 - Prob. 12AYUCh. 7.8 - Prob. 13AYUCh. 7.8 - Prob. 14AYUCh. 7.8 - Prob. 15AYUCh. 7.8 - Prob. 16AYUCh. 7.8 - Prob. 17AYUCh. 7.8 - Prob. 18AYUCh. 7.8 - Prob. 19AYUCh. 7.8 - Prob. 20AYUCh. 7.8 - Prob. 21AYUCh. 7.8 - Prob. 22AYUCh. 7.8 - Prob. 23AYUCh. 7.8 - Prob. 24AYUCh. 7.8 - Prob. 25AYUCh. 7.8 - Prob. 26AYUCh. 7.8 - Prob. 27AYUCh. 7.8 - Prob. 28AYUCh. 7.8 - Prob. 29AYUCh. 7.8 - Prob. 30AYUCh. 7.8 - Prob. 31AYUCh. 7.8 - Prob. 32AYUCh. 7.8 - Prob. 33AYUCh. 7.8 - Prob. 34AYUCh. 7.8 - Prob. 35AYUCh. 7.8 - Prob. 38AYUCh. 7.8 - Prob. 39AYUCh. 7.8 - Prob. 40AYUCh. 7.8 - Prob. 41AYUCh. 7.8 - Prob. 42AYUCh. 7.8 - Prob. 43AYUCh. 7.8 - Prob. 44AYUCh. 7.8 - Prob. 45AYUCh. 7.8 - Prob. 46AYUCh. 7.8 - Prob. 47AYUCh. 7 - Prob. 1RECh. 7 - Prob. 2RECh. 7 - Prob. 3RECh. 7 - Prob. 4RECh. 7 - Prob. 5RECh. 7 - Prob. 6RECh. 7 - Prob. 7RECh. 7 - Prob. 8RECh. 7 - Prob. 9RECh. 7 - Prob. 10RECh. 7 - Prob. 11RECh. 7 - Prob. 12RECh. 7 - Prob. 13RECh. 7 - Prob. 14RECh. 7 - Prob. 15RECh. 7 - Prob. 16RECh. 7 - Prob. 17RECh. 7 - Prob. 18RECh. 7 - Prob. 19RECh. 7 - Prob. 20RECh. 7 - Prob. 21RECh. 7 - Prob. 22RECh. 7 - Prob. 23RECh. 7 - Prob. 24RECh. 7 - Prob. 25RECh. 7 - Prob. 26RECh. 7 - Prob. 27RECh. 7 - Prob. 28RECh. 7 - Prob. 29RECh. 7 - Prob. 30RECh. 7 - Prob. 31RECh. 7 - Prob. 32RECh. 7 - Prob. 33RECh. 7 - Prob. 34RECh. 7 - Prob. 35RECh. 7 - Prob. 36RECh. 7 - Prob. 37RECh. 7 - Prob. 38RECh. 7 - Prob. 39RECh. 7 - Prob. 40RECh. 7 - Prob. 41RECh. 7 - Prob. 42RECh. 7 - Prob. 43RECh. 7 - Prob. 44RECh. 7 - Prob. 45RECh. 7 - Prob. 46RECh. 7 - Prob. 47RECh. 7 - Prob. 48RECh. 7 - Prob. 49RECh. 7 - Prob. 50RECh. 7 - Prob. 51RECh. 7 - Prob. 52RECh. 7 - Prob. 53RECh. 7 - Prob. 54RECh. 7 - Prob. 55RECh. 7 - Prob. 56RECh. 7 - Prob. 57RECh. 7 - Prob. 58RECh. 7 - Prob. 59RECh. 7 - Prob. 60RECh. 7 - Prob. 61RECh. 7 - Prob. 1CTCh. 7 - Prob. 2CTCh. 7 - Prob. 3CTCh. 7 - Prob. 4CTCh. 7 - Prob. 5CTCh. 7 - Prob. 6CTCh. 7 - Prob. 7CTCh. 7 - Prob. 8CTCh. 7 - Prob. 9CTCh. 7 - Prob. 10CTCh. 7 - Prob. 11CTCh. 7 - Prob. 12CTCh. 7 - Prob. 13CTCh. 7 - Prob. 14CTCh. 7 - Prob. 15CTCh. 7 - Prob. 16CTCh. 7 - Prob. 17CTCh. 7 - Prob. 18CTCh. 7 - Prob. 19CTCh. 7 - Prob. 20CTCh. 7 - Prob. 21CTCh. 7 - Prob. 22CTCh. 7 - Prob. 23CTCh. 7 - Prob. 24CTCh. 7 - Prob. 25CTCh. 7 - Prob. 26CTCh. 7 - Prob. 27CTCh. 7 - Prob. 28CTCh. 7 - Prob. 29CTCh. 7 - Prob. 30CTCh. 7 - Prob. 31CTCh. 7 - Prob. 1CRCh. 7 - Prob. 2CRCh. 7 - Prob. 3CRCh. 7 - Prob. 4CRCh. 7 - Prob. 5CRCh. 7 - Prob. 6CRCh. 7 - Prob. 7CRCh. 7 - Prob. 8CRCh. 7 - Prob. 9CRCh. 7 - Prob. 10CRCh. 7 - Prob. 11CRCh. 7 - Prob. 12CRCh. 7 - Prob. 13CRCh. 7 - Prob. 14CR
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