25. Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
25. Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
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![25. Suppose the coefficient matrix of a system of linear equations
has a pivot position in every row. Explain why the system is
consistent.
26. Suppose the coefficient matrix of a linear system of three
equations in three variables has a pivot in each column.
Explain why the system has a unique solution.
27. Restate the last sentence in Theorem 2 using the concept
of pivot columns: "If a linear system is consistent, then the
solution is unique if and only if
28. What would you have to know about the pivot columns in an
augmented matrix in order to know that the linear system is
consistent and has a unique solution?
29. A system of linear equations with fewer equations than
unknowns is sometimes called an underdetermined system.
Suppose that such a system happens to be consistent. Explain
why there must be an infinite number of solutions.
30. Give an example of an inconsistent underdetermined system
of two equations in three unknowns.
32. Suppose an n x (n + 1) matrix is row reduced to reduced
echelon form. Approximately what fraction of the total num-
ber of operations (flops) is involved in the backward phase of
the reduction when n = 30? when n = 300?1
gri
Suppose experimental data are represented by a set of points
in the plane. An interpolating polynomial for the data is a
16
Der sur 10wG
16000 21 115
15
31. A system of linear equations with more equations than un-
knowns is sometimes called an overdetermined system. Can
can
such a system be consistent? Illustrate your answer with a
specific system of three equations in two unknowns.
X1
x3
x2
The general solution of the
system of equations is the line of
Do
intersection of the two planes.
polynomial whose graph passes through every point. In scientific
work, such a polynomial can be used, for example, to estimate
values between the known data points. Another use is to create
curves for graphical images on a computer screen. One method for
finding an interpolating polynomial is to solve a system of linear
equations.
WEB
eft
TE JESION
34.
33. Find the interpolating polynomial p(t) = ao + a₁t + a₂t²
for the data (1, 12), (2, 15), (3, 16). That is, find ao, a₁, and
a2 such that lan
IS
ao + a₁ (1) + a₂(1)² = 12
ao + a₁ (2) + a₂(2)² = 15
ao + a₁ (3) + a₂(3)² = 16
[M] In a wind tunnel experiment, the force on a projectile
due to air resistance was measured at different velocities:
ODD
Velocity (100 ft/sec)
Force (100 lb)
02
4
6
8
10
0 2.90 14.8 39.6 74.3 119
2017
[!
1.2 Row Reduction and Echelon Forms 23
SOLUTIONS TO PRACTICE PROBLEMS
visi
Find an interpolating polynomial for these data and estimate
the force on the projectile when the projectile is travel-
ing at 750 ft/sec. Use p(t) = ao + a₁t + a₂t² + a3t³ + a4tª
+asts. What happens if you try to use a polynomial of degree
less than 5? (Try a cubic polynomial, for instance.
CHOLAUDA SOTO
TAUGH HOTOSY CA
1. The reduced echelon form of the augmented matrix and the corresponding system
1113101997
are
1 0-8 -3
5 Exercises marked with the symbol [M] are designed to be worked
with the aid of a "Matrix program" (a computer program, such as
MATLAB, Maple, Mathematica, MathCad, or Derive, or a
programmable calculator with matrix capabilities, such as those
manufactured by Texas Instruments or Hewlett-Packard).
w daides
0 1 -1
and
desun
X1
The basic variables are x₁ and x2, and the general solution is
x₁ = -3 + 8x3
X1
x2
x₂ = − 1 + x3
X3 is free d
IS
- 8x3 = -3
X2 - X3 = -1
09:22 3120
Note: It is essential that the general solution describe each variable, with any param-
eters clearly identified. The following statement does not describe the solution:
De
x₁ = −3+ 8x3 dring
X₂ = − 1 + x3
(x3 = 1 + x2 Incorrect solution
This description implies that x2 and x3 are both free, which certainly is not the case.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bfb7105-3383-49f1-9e7c-a22283ce619a%2F38b13a84-2f6c-428d-aaf3-8ebc981a54c1%2F13lnvkh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:25. Suppose the coefficient matrix of a system of linear equations
has a pivot position in every row. Explain why the system is
consistent.
26. Suppose the coefficient matrix of a linear system of three
equations in three variables has a pivot in each column.
Explain why the system has a unique solution.
27. Restate the last sentence in Theorem 2 using the concept
of pivot columns: "If a linear system is consistent, then the
solution is unique if and only if
28. What would you have to know about the pivot columns in an
augmented matrix in order to know that the linear system is
consistent and has a unique solution?
29. A system of linear equations with fewer equations than
unknowns is sometimes called an underdetermined system.
Suppose that such a system happens to be consistent. Explain
why there must be an infinite number of solutions.
30. Give an example of an inconsistent underdetermined system
of two equations in three unknowns.
32. Suppose an n x (n + 1) matrix is row reduced to reduced
echelon form. Approximately what fraction of the total num-
ber of operations (flops) is involved in the backward phase of
the reduction when n = 30? when n = 300?1
gri
Suppose experimental data are represented by a set of points
in the plane. An interpolating polynomial for the data is a
16
Der sur 10wG
16000 21 115
15
31. A system of linear equations with more equations than un-
knowns is sometimes called an overdetermined system. Can
can
such a system be consistent? Illustrate your answer with a
specific system of three equations in two unknowns.
X1
x3
x2
The general solution of the
system of equations is the line of
Do
intersection of the two planes.
polynomial whose graph passes through every point. In scientific
work, such a polynomial can be used, for example, to estimate
values between the known data points. Another use is to create
curves for graphical images on a computer screen. One method for
finding an interpolating polynomial is to solve a system of linear
equations.
WEB
eft
TE JESION
34.
33. Find the interpolating polynomial p(t) = ao + a₁t + a₂t²
for the data (1, 12), (2, 15), (3, 16). That is, find ao, a₁, and
a2 such that lan
IS
ao + a₁ (1) + a₂(1)² = 12
ao + a₁ (2) + a₂(2)² = 15
ao + a₁ (3) + a₂(3)² = 16
[M] In a wind tunnel experiment, the force on a projectile
due to air resistance was measured at different velocities:
ODD
Velocity (100 ft/sec)
Force (100 lb)
02
4
6
8
10
0 2.90 14.8 39.6 74.3 119
2017
[!
1.2 Row Reduction and Echelon Forms 23
SOLUTIONS TO PRACTICE PROBLEMS
visi
Find an interpolating polynomial for these data and estimate
the force on the projectile when the projectile is travel-
ing at 750 ft/sec. Use p(t) = ao + a₁t + a₂t² + a3t³ + a4tª
+asts. What happens if you try to use a polynomial of degree
less than 5? (Try a cubic polynomial, for instance.
CHOLAUDA SOTO
TAUGH HOTOSY CA
1. The reduced echelon form of the augmented matrix and the corresponding system
1113101997
are
1 0-8 -3
5 Exercises marked with the symbol [M] are designed to be worked
with the aid of a "Matrix program" (a computer program, such as
MATLAB, Maple, Mathematica, MathCad, or Derive, or a
programmable calculator with matrix capabilities, such as those
manufactured by Texas Instruments or Hewlett-Packard).
w daides
0 1 -1
and
desun
X1
The basic variables are x₁ and x2, and the general solution is
x₁ = -3 + 8x3
X1
x2
x₂ = − 1 + x3
X3 is free d
IS
- 8x3 = -3
X2 - X3 = -1
09:22 3120
Note: It is essential that the general solution describe each variable, with any param-
eters clearly identified. The following statement does not describe the solution:
De
x₁ = −3+ 8x3 dring
X₂ = − 1 + x3
(x3 = 1 + x2 Incorrect solution
This description implies that x2 and x3 are both free, which certainly is not the case.
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