EBK ALGEBRA FOUNDATIONS
15th Edition
ISBN: 9780321978929
Author: Martin-Gay
Publisher: PEARSON
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Chapter 11.1, Problem 66ES
To determine
To draw a graph of two linear equations whose associated system has infinite number ofsolutions
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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]
part b please
Chapter 11 Solutions
EBK ALGEBRA FOUNDATIONS
Ch. 11.1 - Prob. 1VRVCCh. 11.1 - Prob. 2VRVCCh. 11.1 - Prob. 3VRVCCh. 11.1 - Prob. 4VRVCCh. 11.1 - Prob. 5VRVCCh. 11.1 - Prob. 6VRVCCh. 11.1 - Prob. 1ESCh. 11.1 - Prob. 2ESCh. 11.1 - Prob. 3ESCh. 11.1 - Prob. 4ES
Ch. 11.1 - Prob. 5ESCh. 11.1 - Prob. 6ESCh. 11.1 - Prob. 7ESCh. 11.1 - Prob. 8ESCh. 11.1 - Prob. 9ESCh. 11.1 - Objective A Determine whether each ordered pair is...Ch. 11.1 - Prob. 11ESCh. 11.1 - Prob. 12ESCh. 11.1 - Prob. 13ESCh. 11.1 - Prob. 14ESCh. 11.1 - Prob. 15ESCh. 11.1 - Prob. 16ESCh. 11.1 - Prob. 17ESCh. 11.1 - Prob. 18ESCh. 11.1 - Prob. 19ESCh. 11.1 - Prob. 20ESCh. 11.1 - Prob. 21ESCh. 11.1 - Prob. 22ESCh. 11.1 - Prob. 23ESCh. 11.1 - Prob. 24ESCh. 11.1 - Prob. 25ESCh. 11.1 - Prob. 26ESCh. 11.1 - Prob. 27ESCh. 11.1 - Prob. 28ESCh. 11.1 - Prob. 29ESCh. 11.1 - Prob. 30ESCh. 11.1 - Prob. 31ESCh. 11.1 - Prob. 32ESCh. 11.1 - Prob. 33ESCh. 11.1 - Prob. 34ESCh. 11.1 - Prob. 35ESCh. 11.1 - Prob. 36ESCh. 11.1 - Prob. 37ESCh. 11.1 - Prob. 38ESCh. 11.1 - Prob. 39ESCh. 11.1 - Prob. 40ESCh. 11.1 - Prob. 41ESCh. 11.1 - Prob. 42ESCh. 11.1 - Prob. 43ESCh. 11.1 - Prob. 44ESCh. 11.1 - Prob. 45ESCh. 11.1 - Prob. 46ESCh. 11.1 - Prob. 47ESCh. 11.1 - Prob. 48ESCh. 11.1 - Prob. 49ESCh. 11.1 - Prob. 50ESCh. 11.1 - Prob. 51ESCh. 11.1 - Prob. 52ESCh. 11.1 - Prob. 53ESCh. 11.1 - Prob. 54ESCh. 11.1 - Prob. 55ESCh. 11.1 - Prob. 56ESCh. 11.1 - Prob. 57ESCh. 11.1 - Prob. 58ESCh. 11.1 - Prob. 59ESCh. 11.1 - Prob. 60ESCh. 11.1 - Prob. 61ESCh. 11.1 - Prob. 62ESCh. 11.1 - Prob. 63ESCh. 11.1 - Prob. 64ESCh. 11.1 - Prob. 65ESCh. 11.1 - Prob. 66ESCh. 11.1 - Prob. 67ESCh. 11.1 - Prob. 68ESCh. 11.1 - Prob. 69ESCh. 11.1 - Prob. 70ESCh. 11.1 - Prob. 71ESCh. 11.1 - Prob. 72ESCh. 11.1 - Prob. 73ESCh. 11.1 - Prob. 74ESCh. 11.1 - Below are tables of values for two linear...Ch. 11.1 - Prob. 76ESCh. 11.1 - Prob. 77ESCh. 11.1 - Prob. 78ESCh. 11.2 - Prob. 1VRVCCh. 11.2 - Prob. 2VRVCCh. 11.2 - Prob. 3VRVCCh. 11.2 - Prob. 4VRVCCh. 11.2 - Prob. 5VRVCCh. 11.2 - Prob. 6VRVCCh. 11.2 - Prob. 1ESCh. 11.2 - Prob. 2ESCh. 11.2 - Prob. 3ESCh. 11.2 - Prob. 4ESCh. 11.2 - Prob. 5ESCh. 11.2 - Prob. 6ESCh. 11.2 - Prob. 7ESCh. 11.2 - Prob. 8ESCh. 11.2 - Prob. 9ESCh. 11.2 - Prob. 10ESCh. 11.2 - Prob. 11ESCh. 11.2 - Prob. 12ESCh. 11.2 - Prob. 13ESCh. 11.2 - Prob. 14ESCh. 11.2 - Prob. 15ESCh. 11.2 - Prob. 16ESCh. 11.2 - Prob. 17ESCh. 11.2 - Prob. 18ESCh. 11.2 - Prob. 19ESCh. 11.2 - Prob. 20ESCh. 11.2 - Solve each system of equations by the substitution...Ch. 11.2 - Solve each system of equations by the substitution...Ch. 11.2 - Prob. 23ESCh. 11.2 - Prob. 24ESCh. 11.2 - Prob. 25ESCh. 11.2 - Prob. 26ESCh. 11.2 - Prob. 27ESCh. 11.2 - Prob. 28ESCh. 11.2 - Prob. 29ESCh. 11.2 - Prob. 30ESCh. 11.2 - Prob. 31ESCh. 11.2 - Prob. 32ESCh. 11.2 - Prob. 33ESCh. 11.2 - Prob. 34ESCh. 11.2 - Prob. 35ESCh. 11.2 - Prob. 36ESCh. 11.2 - Prob. 37ESCh. 11.2 - Prob. 38ESCh. 11.2 - Prob. 39ESCh. 11.2 - Prob. 40ESCh. 11.2 - Prob. 41ESCh. 11.2 - Prob. 42ESCh. 11.2 - Prob. 43ESCh. 11.2 - Prob. 44ESCh. 11.2 - Prob. 45ESCh. 11.2 - Prob. 46ESCh. 11.2 - Prob. 47ESCh. 11.2 - Prob. 48ESCh. 11.2 - Prob. 49ESCh. 11.2 - Prob. 50ESCh. 11.2 - Prob. 51ESCh. 11.2 - Prob. 52ESCh. 11.2 - Prob. 53ESCh. 11.2 - Prob. 54ESCh. 11.2 - Solve each system by substitution. When necessary,...Ch. 11.2 - Prob. 56ESCh. 11.2 - Prob. 57ESCh. 11.2 - Prob. 58ESCh. 11.3 - Prob. 1VRVCCh. 11.3 - Prob. 2VRVCCh. 11.3 - Prob. 3VRVCCh. 11.3 - Prob. 4VRVCCh. 11.3 - Prob. 1ESCh. 11.3 - Prob. 2ESCh. 11.3 - Prob. 3ESCh. 11.3 - Prob. 4ESCh. 11.3 - Prob. 5ESCh. 11.3 - Prob. 6ESCh. 11.3 - Prob. 7ESCh. 11.3 - Prob. 8ESCh. 11.3 - Prob. 9ESCh. 11.3 - Prob. 10ESCh. 11.3 - Prob. 11ESCh. 11.3 - Prob. 12ESCh. 11.3 - Prob. 13ESCh. 11.3 - Prob. 14ESCh. 11.3 - Prob. 15ESCh. 11.3 - Prob. 16ESCh. 11.3 - Prob. 17ESCh. 11.3 - Prob. 18ESCh. 11.3 - Prob. 19ESCh. 11.3 - Prob. 20ESCh. 11.3 - Prob. 21ESCh. 11.3 - Prob. 22ESCh. 11.3 - Prob. 23ESCh. 11.3 - Prob. 24ESCh. 11.3 - Prob. 25ESCh. 11.3 - Prob. 26ESCh. 11.3 - Prob. 27ESCh. 11.3 - Prob. 28ESCh. 11.3 - Prob. 29ESCh. 11.3 - Prob. 30ESCh. 11.3 - Prob. 31ESCh. 11.3 - Prob. 32ESCh. 11.3 - Prob. 33ESCh. 11.3 - Prob. 34ESCh. 11.3 - Prob. 35ESCh. 11.3 - Prob. 36ESCh. 11.3 - Prob. 37ESCh. 11.3 - Prob. 38ESCh. 11.3 - Prob. 39ESCh. 11.3 - Prob. 40ESCh. 11.3 - Prob. 41ESCh. 11.3 - Prob. 42ESCh. 11.3 - Prob. 43ESCh. 11.3 - Prob. 44ESCh. 11.3 - Prob. 45ESCh. 11.3 - Prob. 46ESCh. 11.3 - Prob. 47ESCh. 11.3 - Prob. 48ESCh. 11.3 - Prob. 49ESCh. 11.3 - Prob. 50ESCh. 11.3 - Prob. 51ESCh. 11.3 - Prob. 52ESCh. 11.3 - Prob. 53ESCh. 11.3 - Prob. 54ESCh. 11.3 - Prob. 55ESCh. 11.3 - Prob. 56ESCh. 11.3 - Prob. 57ESCh. 11.3 - Prob. 58ESCh. 11.3 - Prob. 59ESCh. 11.3 - Prob. 60ESCh. 11.3 - Prob. 61ESCh. 11.3 - Prob. 62ESCh. 11.3 - Prob. 63ESCh. 11.3 - Prob. 64ESCh. 11.3 - Prob. 65ESCh. 11.3 - Prob. 66ESCh. 11.3 - Prob. 67ESCh. 11.3 - Prob. 68ESCh. 11.3 - Prob. 69ESCh. 11.3 - Prob. 70ESCh. 11.3 - Prob. 71ESCh. 11.3 - Prob. 72ESCh. 11.3 - Prob. 73ESCh. 11.3 - Prob. 74ESCh. 11.4 - Prob. 1VRVCCh. 11.4 - Prob. 2VRVCCh. 11.4 - Prob. 3VRVCCh. 11.4 - Prob. 4VRVCCh. 11.4 - Prob. 1ESCh. 11.4 - Prob. 2ESCh. 11.4 - Prob. 3ESCh. 11.4 - Prob. 4ESCh. 11.4 - Prob. 5ESCh. 11.4 - Prob. 6ESCh. 11.4 - Prob. 7ESCh. 11.4 - Prob. 8ESCh. 11.4 - Prob. 9ESCh. 11.4 - Prob. 10ESCh. 11.4 - Prob. 11ESCh. 11.4 - Prob. 12ESCh. 11.4 - Prob. 13ESCh. 11.4 - Prob. 14ESCh. 11.4 - Prob. 15ESCh. 11.4 - Prob. 16ESCh. 11.4 - Solve each system. See Examples 1 through 5. 17....Ch. 11.4 - Prob. 18ESCh. 11.4 - Prob. 19ESCh. 11.4 - Prob. 20ESCh. 11.4 - Prob. 21ESCh. 11.4 - Prob. 22ESCh. 11.4 - Prob. 23ESCh. 11.4 - Prob. 24ESCh. 11.4 - Prob. 25ESCh. 11.4 - Prob. 26ESCh. 11.4 - Prob. 27ESCh. 11.4 - Prob. 28ESCh. 11.4 - Prob. 29ESCh. 11.4 - Prob. 30ESCh. 11.4 - Prob. 31ESCh. 11.4 - Prob. 32ESCh. 11.4 - Prob. 33ESCh. 11.4 - Prob. 34ESCh. 11.4 - Prob. 35ESCh. 11.4 - Prob. 36ESCh. 11.4 - Prob. 37ESCh. 11.4 - Prob. 38ESCh. 11.4 - Prob. 39ESCh. 11.4 - Prob. 40ESCh. 11.4 - Prob. 41ESCh. 11.4 - Prob. 42ESCh. 11.4 - Prob. 43ESCh. 11.4 - Prob. 44ESCh. 11.4 - Prob. 45ESCh. 11.4 - Prob. 46ESCh. 11.5 - Prob. 1ESCh. 11.5 - Prob. 2ESCh. 11.5 - Prob. 3ESCh. 11.5 - Prob. 4ESCh. 11.5 - Prob. 5ESCh. 11.5 - Prob. 6ESCh. 11.5 - Prob. 7ESCh. 11.5 - Prob. 8ESCh. 11.5 - Prob. 9ESCh. 11.5 - Prob. 10ESCh. 11.5 - Prob. 11ESCh. 11.5 - Prob. 12ESCh. 11.5 - Prob. 13ESCh. 11.5 - Prob. 14ESCh. 11.5 - Prob. 15ESCh. 11.5 - Prob. 16ESCh. 11.5 - Prob. 17ESCh. 11.5 - Prob. 18ESCh. 11.5 - Prob. 19ESCh. 11.5 - Prob. 20ESCh. 11.5 - Prob. 21ESCh. 11.5 - Prob. 22ESCh. 11.5 - Prob. 23ESCh. 11.5 - Prob. 24ESCh. 11.5 - Prob. 25ESCh. 11.5 - Prob. 26ESCh. 11.5 - Prob. 27ESCh. 11.5 - Prob. 28ESCh. 11.5 - Prob. 29ESCh. 11.5 - Prob. 30ESCh. 11.5 - Prob. 31ESCh. 11.5 - Prob. 32ESCh. 11.5 - Prob. 33ESCh. 11.5 - Prob. 34ESCh. 11.5 - Prob. 35ESCh. 11.5 - Prob. 36ESCh. 11.5 - Prob. 37ESCh. 11.5 - Prob. 38ESCh. 11.5 - Prob. 39ESCh. 11.5 - Prob. 40ESCh. 11.5 - Prob. 41ESCh. 11.5 - A charily fund-raiser consisted of a spaghetti...Ch. 11.5 - Prob. 43ESCh. 11.5 - Prob. 44ESCh. 11.5 - Prob. 45ESCh. 11.5 - Prob. 46ESCh. 11.5 - Prob. 47ESCh. 11.5 - Prob. 48ESCh. 11.5 - Prob. 49ESCh. 11.5 - Prob. 50ESCh. 11.5 - Prob. 51ESCh. 11.5 - Prob. 52ESCh. 11.5 - Prob. 53ESCh. 11.5 - Prob. 54ESCh. 11.5 - Prob. 55ESCh. 11.5 - Prob. 56ESCh. 11.5 - Objective B Given the cost function C(x) and the...Ch. 11.5 - Objective B Given the cost function C(x) and the...Ch. 11.5 - Prob. 59ESCh. 11.5 - Prob. 60ESCh. 11.5 - Prob. 61ESCh. 11.5 - Prob. 62ESCh. 11.5 - Prob. 63ESCh. 11.5 - Prob. 64ESCh. 11.5 - Prob. 65ESCh. 11.5 - Prob. 66ESCh. 11.5 - Prob. 67ESCh. 11.5 - Prob. 68ESCh. 11.5 - Prob. 69ESCh. 11.5 - Prob. 70ESCh. 11.5 - Prob. 71ESCh. 11.5 - Prob. 72ESCh. 11.5 - Prob. 73ESCh. 11.5 - Prob. 74ESCh. 11.5 - Prob. 75ESCh. 11.5 - Prob. 76ESCh. 11.5 - Prob. 77ESCh. 11.5 - Prob. 78ESCh. 11.5 - Prob. 79ESCh. 11.5 - Prob. 80ESCh. 11.5 - Prob. 81ESCh. 11.5 - Prob. 82ESCh. 11.5 - Prob. 83ESCh. 11.5 - Prob. 84ESCh. 11.5 - Prob. 85ESCh. 11.5 - Prob. 86ESCh. 11 - Solve each system by either the addition method or...Ch. 11 - Prob. 2IRCh. 11 - Prob. 3IRCh. 11 - Prob. 4IRCh. 11 - Prob. 5IRCh. 11 - Prob. 6IRCh. 11 - Prob. 7IRCh. 11 - Prob. 8IRCh. 11 - Prob. 9IRCh. 11 - Prob. 10IRCh. 11 - Prob. 11IRCh. 11 - Prob. 12IRCh. 11 - Prob. 13IRCh. 11 - Prob. 14IRCh. 11 - Prob. 15IRCh. 11 - Prob. 16IRCh. 11 - Prob. 17IRCh. 11 - Prob. 18IRCh. 11 - Prob. 19IRCh. 11 - Prob. 20IRCh. 11 - Prob. 21IRCh. 11 - Prob. 22IRCh. 11 - Prob. 1VCCh. 11 - Prob. 2VCCh. 11 - Prob. 3VCCh. 11 - Prob. 4VCCh. 11 - Prob. 5VCCh. 11 - Prob. 6VCCh. 11 - Prob. 7VCCh. 11 - Prob. 1RCh. 11 - Prob. 2RCh. 11 - Prob. 3RCh. 11 - Prob. 4RCh. 11 - Prob. 5RCh. 11 - Prob. 6RCh. 11 - Prob. 7RCh. 11 - Prob. 8RCh. 11 - Prob. 9RCh. 11 - Prob. 10RCh. 11 - Prob. 11RCh. 11 - Prob. 12RCh. 11 - Prob. 13RCh. 11 - Prob. 14RCh. 11 - Prob. 15RCh. 11 - Prob. 16RCh. 11 - Prob. 17RCh. 11 - Prob. 18RCh. 11 - Prob. 19RCh. 11 - Prob. 20RCh. 11 - Prob. 21RCh. 11 - Solve each system of equations by the addition...Ch. 11 - Prob. 23RCh. 11 - Prob. 24RCh. 11 - Prob. 25RCh. 11 - Prob. 26RCh. 11 - Prob. 27RCh. 11 - Prob. 28RCh. 11 - Prob. 29RCh. 11 - Prob. 30RCh. 11 - Prob. 31RCh. 11 - Prob. 32RCh. 11 - Prob. 33RCh. 11 - Prob. 34RCh. 11 - Prob. 35RCh. 11 - Prob. 36RCh. 11 - Prob. 37RCh. 11 - Prob. 38RCh. 11 - Prob. 39RCh. 11 - Prob. 40RCh. 11 - Prob. 41RCh. 11 - Prob. 42RCh. 11 - Prob. 43RCh. 11 - Solve each problem by writing and solving a system...Ch. 11 - Prob. 45RCh. 11 - Prob. 46RCh. 11 - Prob. 47RCh. 11 - Prob. 48RCh. 11 - Prob. 49RCh. 11 - Prob. 50RCh. 11 - Prob. 51RCh. 11 - Prob. 52RCh. 11 - Prob. 53RCh. 11 - Prob. 54RCh. 11 - Prob. 55RCh. 11 - Prob. 56RCh. 11 - Prob. 57RCh. 11 - Prob. 58RCh. 11 - Prob. 59RCh. 11 - Prob. 60RCh. 11 - Solve each problem by writing and solving a system...Ch. 11 - Prob. 62RCh. 11 - Prob. 63RCh. 11 - Prob. 1TCh. 11 - Prob. 2TCh. 11 - Prob. 3TCh. 11 - Prob. 4TCh. 11 - Prob. 5TCh. 11 - Prob. 6TCh. 11 - Prob. 7TCh. 11 - Prob. 8TCh. 11 - Prob. 9TCh. 11 - Prob. 10TCh. 11 - Prob. 11TCh. 11 - Prob. 12TCh. 11 - Prob. 13TCh. 11 - Prob. 14TCh. 11 - Prob. 15TCh. 11 - Prob. 16TCh. 11 - Prob. 17TCh. 11 - Prob. 18TCh. 11 - Prob. 19TCh. 11 - Prob. 20TCh. 11 - Prob. 21TCh. 11 - Prob. 1CRCh. 11 - Prob. 2CRCh. 11 - Prob. 3CRCh. 11 - Prob. 4CRCh. 11 - Prob. 5CRCh. 11 - Prob. 6CRCh. 11 - Prob. 7CRCh. 11 - Prob. 8CRCh. 11 - Prob. 9CRCh. 11 - Prob. 10CRCh. 11 - Prob. 11CRCh. 11 - Prob. 12CRCh. 11 - Prob. 13CRCh. 11 - Prob. 14CRCh. 11 - Prob. 15CRCh. 11 - Prob. 16CRCh. 11 - Prob. 17CRCh. 11 - Solve P=a+b+c for b.Ch. 11 - Prob. 19CRCh. 11 - Prob. 20CRCh. 11 - Prob. 21CRCh. 11 - Prob. 22CRCh. 11 - Prob. 23CRCh. 11 - Prob. 24CRCh. 11 - Prob. 25CRCh. 11 - Prob. 26CRCh. 11 - Prob. 27CRCh. 11 - Prob. 28CRCh. 11 - Prob. 29CRCh. 11 - Prob. 30CRCh. 11 - Prob. 31CRCh. 11 - Prob. 32CRCh. 11 - Prob. 33CRCh. 11 - Prob. 34CRCh. 11 - Prob. 35CRCh. 11 - Prob. 36CRCh. 11 - Solve the system. 37. {x+2y=72x+2y=13Ch. 11 - Prob. 38CRCh. 11 - Prob. 39CRCh. 11 - Prob. 40CRCh. 11 - Prob. 41CRCh. 11 - Prob. 42CRCh. 11 - Prob. 43CRCh. 11 - Prob. 44CR
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