Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
5th Edition
ISBN: 9780134689531
Author: Lee Johnson, Dean Riess, Jimmy Arnold
Publisher: PEARSON
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Chapter 7.1, Problem 7E
In Exercises
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Find an orthogonal change of variables, x = Py, that transforms the quadratic form
q(x) = x² + x²+2x1x2 − 2x2x3
into a quadratic form no cross-product form. Write the new quadratic form.
Consider the following quadratic form on R²:
- 4608x₁² + 2/2/²
2 29
-X1
17
17
Q(x):
=
b)
x₁
2
a) Determine whether Q is positive or negative (semi)definite.
b) Find a change of variables x = Py that converts Q to a form with no cross-product terms.
and y2.
c) write Q in terms of the new variables y1
In your formula, write 'y_1' and 'y_2' for y₁ and y2. Use the square root symbol '√' where needed to
give an exact value for your answer.
y1
0012
+400/Xx1
a) The quadratic form is
c) Q(x) = 0
-x₂² +-
-x1x2
17
Positive Definite
Negative Definite
Positive Semidefinite
Negative Semidefinite
Indefinite
Find an orthogonal change of variables that eliminates the cross product terms in the quadratic form Q, and express Q in terms of
the new variables.
7x구 +6x2 + 5x금-4r x2 + 4x2.13
2
A substitution x = Py that eliminates cross-product terms is xį = -y1 +Y2 - V3, x2 = -z1 + zy2 +zV3,
2
1
X3 = -V1 + V2 – V3. The new quadratic form is 3y – 6y + 9y?.
O A substitution x = Py that eliminates cross-product terms is x1 = - y1+ 2y2 - 2y3, X2 = - 2y1+ y2+ 2y3, X3 = 2y1+2y2 + y3-
The new quadratic form is 3y + 6y + 9y.
2
1
+
2
2
A substitution x = Py that eliminates cross-product terms is x = -
2
X2 =
1
+
+
1
V2 + zV3. The new quadratic form is 3y + 6y + 9y.
X3 =
1
2
2
2
A substitution x = Py that eliminates cross-product terms is x = -I -2 -y3, x2 = -1 - 32 + 33,
2
1
X3 = 7Y1 + zV2 + V3. The new quadratic form is 6y+ 5y + 3y.
O A substitution x = Py that eliminates cross-product terms is x1 = - y1- 2y2 - 2y3, X2 = - 2y1 – Y2+ 2y3, X3 = 2y1+2y2 + y3-
The new quadratic form is 9y + 3y + 6y.
Chapter 7 Solutions
Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 7.1 - In Exercises 16, find a symmetric matrix A such...Ch. 7.1 - Prob. 2ECh. 7.1 - In Exercises 16, find a symmetric matrix A such...Ch. 7.1 - Prob. 4ECh. 7.1 - In Exercises 16, find a symmetric matrix A such...Ch. 7.1 - Prob. 6ECh. 7.1 - In Exercises 712, find a substitution x=Qy that...Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...
Ch. 7.1 - In Exercises 712, find a substitution x=Qy that...Ch. 7.1 - Prob. 12ECh. 7.1 - Prob. 13ECh. 7.1 - In Exercises 1320, find a substitution x=Qy where...Ch. 7.1 - In Exercises 1320, find a substitution x=Qy where...Ch. 7.1 - Prob. 16ECh. 7.1 - In Exercises 1320, find a substitution x=Qy where...Ch. 7.1 - Prob. 18ECh. 7.1 - In Exercises 1320, find a substitution x=Qy where...Ch. 7.1 - In Exercises 1320, find a substitution x=Qy where...Ch. 7.1 - Prob. 21ECh. 7.1 - Prob. 22ECh. 7.1 - Prove property b of Theorem 2. THEOREM 2 Let q(x)...Ch. 7.1 - Prob. 24ECh. 7.1 - Prob. 25ECh. 7.1 - Let A be an (nn) symmetric matrix and consider the...Ch. 7.1 - Prob. 27ECh. 7.1 - Let A be an (nn) symmetric matrix, and let S be an...Ch. 7.2 - Prob. 1ECh. 7.2 - Prob. 2ECh. 7.2 - Prob. 3ECh. 7.2 - Prob. 4ECh. 7.2 - Prob. 5ECh. 7.2 - Prob. 6ECh. 7.2 - Prob. 7ECh. 7.2 - Prob. 8ECh. 7.2 - Prob. 9ECh. 7.2 - Prob. 10ECh. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - In Exercise 1-10, reduce the given matrix to...Ch. 7.3 - Prob. 11ECh. 7.3 - Prob. 12ECh. 7.3 - Prob. 13ECh. 7.3 - Prob. 14ECh. 7.3 - Prob. 15ECh. 7.3 - Exercise 1522 deal with permutation matrices....Ch. 7.3 - Prob. 17ECh. 7.3 - Exercise 1522 deal with permutation matrices....Ch. 7.3 - Prob. 19ECh. 7.3 - Prob. 20ECh. 7.3 - Prob. 21ECh. 7.3 - Prob. 22ECh. 7.4 - Prob. 1ECh. 7.4 - Prob. 2ECh. 7.4 - Prob. 3ECh. 7.4 - Prob. 4ECh. 7.4 - Prob. 5ECh. 7.4 - Prob. 6ECh. 7.4 - Prob. 7ECh. 7.4 - Prob. 8ECh. 7.4 - Prob. 9ECh. 7.4 - Prob. 10ECh. 7.4 - Prob. 11ECh. 7.4 - Prob. 12ECh. 7.4 - Prob. 13ECh. 7.4 - Prob. 14ECh. 7.4 - Prob. 15ECh. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - Prob. 19ECh. 7.4 - Prob. 20ECh. 7.4 - Prob. 21ECh. 7.4 - Prob. 22ECh. 7.4 - Prob. 23ECh. 7.5 - Let Q=IbuuT be the Householder matrix defined by...Ch. 7.5 - Prob. 2ECh. 7.5 - Let Q=IbuuT be the Householder matrix defined by...Ch. 7.5 - Prob. 4ECh. 7.5 - Prob. 5ECh. 7.5 - Prob. 6ECh. 7.5 - Prob. 7ECh. 7.5 - Prob. 8ECh. 7.5 - For the given vectors v and w in Exercise 9-14,...Ch. 7.5 - Prob. 10ECh. 7.5 - Prob. 11ECh. 7.5 - For the given vectors v and w in Exercise 9-14,...Ch. 7.5 - Prob. 13ECh. 7.5 - Prob. 14ECh. 7.5 - Prob. 15ECh. 7.5 - Prob. 16ECh. 7.5 - Prob. 17ECh. 7.5 - In Exercises 15-20, find a Householder matrix Q...Ch. 7.5 - Prob. 19ECh. 7.5 - Prob. 20ECh. 7.5 - Prob. 21ECh. 7.5 - Prob. 22ECh. 7.5 - Consider the (nn) Householder matrix Q=IbuuT,...Ch. 7.5 - Prob. 24ECh. 7.5 - Consider a (44) matrix B of the form shown in (9),...Ch. 7.6 - Prob. 1ECh. 7.6 - Prob. 2ECh. 7.6 - Prob. 3ECh. 7.6 - Prob. 4ECh. 7.6 - Prob. 5ECh. 7.6 - Prob. 6ECh. 7.6 - Prob. 7ECh. 7.6 - Prob. 8ECh. 7.6 - Prob. 9ECh. 7.6 - Prob. 10ECh. 7.6 - Prob. 11ECh. 7.6 - Prob. 12ECh. 7.6 - Prob. 13ECh. 7.6 - Prob. 14ECh. 7.6 - Prob. 15ECh. 7.6 - Prob. 16ECh. 7.6 - Prob. 17ECh. 7.6 - Prob. 18ECh. 7.6 - Prob. 19ECh. 7.7 - Prob. 1ECh. 7.7 - Prob. 2ECh. 7.7 - Prob. 3ECh. 7.7 - Prob. 4ECh. 7.7 - Prob. 5ECh. 7.7 - Prob. 6ECh. 7.7 - Prob. 7ECh. 7.7 - Exercise 6 shows that eigenvectors of a symmetric...Ch. 7.8 - Find a full set of eigenvectors and generalized...Ch. 7.8 - Find a full set of eigenvectors and generalized...Ch. 7.8 - Solve x=Ax, x(0)=x0 by transforming A to...Ch. 7.8 - Prob. 4ECh. 7.8 - Prob. 5ECh. 7.8 - Prob. 6ECh. 7.8 - Prob. 7ECh. 7.8 - Prob. 8ECh. 7.SE - Prob. 1SECh. 7.SE - Prob. 2SECh. 7.SE - Prob. 3SECh. 7.SE - Prob. 4SECh. 7.SE - Prob. 5SECh. 7.CE - Let A be a (33) nonsingular matrix. Use the...Ch. 7.CE - Let A and B be similar (nn) matrices and let p(t)...Ch. 7.CE - Prob. 3CECh. 7.CE - Let A be a (33) matrix. a Use the Cayley-Hamilton...
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- Please do botharrow_forwardConsider the quadratic form q = 30x² + (−5)x² + (−120)x1x2. Determine a change of variables (make sure the coefficient of ₁ is positive) x₁+ X2 Y1 and Y2 = so that q y² x₁+ I2 y. (The coefficient of y² is negative.)arrow_forwardConsider the following quadratic form on R²: 43 2 57 2 + -x7² 25 -X₂ 2 25 Q(x) = [₁] = 48 a) Determine whether Q is positive or negative (semi)definite. b) Find a change of variables x = Py that converts Q to a form with no cross-product terms. c) write Q in terms of the new variables y 1 and y2. In your formula, write 'y_1' and 'y_2' for y₁ and y2. Use the square root symbol 'V' where needed to give an exact value for your answer. [00₁] 0 0 1 00 42 +. a) The quadratic form is c) Q(x) = 0 25 -X1X2 Positive Definite Negative Definite Positive Semidefinite Negative Semidefinite Indefinitearrow_forward
- Why W can't be zero?arrow_forward1. Determine the characteristic of the roots of the quadratic equations. a) x? – x – 30 = 0 b) 7x – 3x – 7 = 0 | c) x? + 8x – 9 = 0 d) 8x +4 –x? = 0arrow_forwardThe conditions to test the positivity of symetric quadratic form of four variables are f(1,0,0) 2 0, f (1, 1, 1) > 0 True Falsearrow_forward
- How do I explan that x2+4xy-Sin(y2) is not a quadratic form?arrow_forwardMake a change of variable, x = Py, that transforms the quadratic form x? + 10xjx2 +x; into a quadratic form with no cross-product term. Give P and the new quadratic form.arrow_forwardMake a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product term. Give Pand the new quadratic form. Q(x) = 3x, + 6x, + 4x, X2 A. 1 2 1 2 V5 V5 - 7y - 2y3 V5 7y +2y? V5 P = 1 1 V5 V5 V5 V5 D. 1 P = 2 - 2 7y1 +2y2 1 1 P= - 2 :7y주 + 2y2 1 P. B.arrow_forward
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