In Exercises 19 and 20, choose h and k such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part.
In Exercises 19 and 20, choose h and k such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part.
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CHAPTER 1
Linear Equations in Linear Algebra
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d. Whenever a system has free variables, the solution set con-
tains many solutions.
e. Ageneral solution of a system is an explicit description of
all solutions of the system.
16. a.
b.
23. Suppose a 3x 5 coefficient matrix for a system has three pivot
columns. Is the system consistent? Why or why not?
In Exercises 17 and 18, determine the value(s) of h such that the
matrix is the augmented matrix of a consistent linear system.
24. Suppose a system of linear equations has a 3 x5 augmented
matrix whose fifth column is a pivot column. Is the system
consistent? Why (or why not)?
[2 3 h]
25. Suppose the coefficient matrix of a system of linear equations
has a pivot position in every row. Explain why the system is
1 -3
17.
18.
consistent.
In Exercises 19 and 20, choose h and k such that the system has (a)
no solution, (b) a unique solution, and (c) many solutions. Give
separate answers for each part.
26. Suppose the coefficient matrix of a linear system of three equa-
tions 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 so-
lution is unique if and only if.
19. x + hx2 = 2
20. xi + 3x2 2
4x1 + 8x2 = k
3x1 + hx2 = k
In Exercises 21 and 22, mark each statement True or False. Justify
each answer.
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 un-
knowns is sometimes called an underdetermined system. Sup-
pose that such a system happens to be consistent. Explain why
there must be an infinite number of solutions.
21. a. In some cases, a matrix may be row reduced to more than
one matrix in reduced echelon form, using different se-
quences of row operations.
b. The row reduction algorithm applies only to augmented
matrices for a linear system.
c. A basic variable in a linear system is a variable that corre-
sponds to a pivot column in the coefficient matrix.
30. Give an example of an inconsistent underdetermined system
of two equations in three unknowns.
31. A system of linear equations with more equations than un-
knowns is sometimes called an overdetermined system. Can
such a system be consistent? Illustrate your answer with a
specific system of three equations in two unknowns.
d. Finding a parametric description of the solution set of a
linear system is the same as solving the system.
e. If one row in an echelon form of an augmented matrix
is [0 0 0 5 0], then the associated linear system is
inconsistent.
32. Suppose an n x(n + 1) matrix is row reduced to reduced ech-
elon form. Approximately what fraction of the total number
of operations (flops) is involved in the backward phase of the
reduction when n = 30? when n = 300?
22. a. The echelon form of a matrix is unique.
b. The pivot positions in a matrix depend on whether row
interchanges are used in the row reduction process.
Suppose experimental data are represented by a set of points in
c. Reducing a matrix to echelon form is called the forward the plane. An interpolating polynomial for the data is a polyno-
mial 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.
phase of the row reduction process.
100% +
*True/false questions of this type will appear in many sections. Methods
for justifying your answers were described before Exercises 23 and 24 in
Section 1.1.
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26
CHAPTER 1
Linear Equations in Linear Algebra
n Shot
.17.41 PM
d. Whenever a system has free variables, the solution set con-
tains many solutions.
e. Ageneral solution of a system is an explicit description of
all solutions of the system.
16. a.
b.
23. Suppose a 3x 5 coefficient matrix for a system has three pivot
columns. Is the system consistent? Why or why not?
In Exercises 17 and 18, determine the value(s) of h such that the
matrix is the augmented matrix of a consistent linear system.
24. Suppose a system of linear equations has a 3 x5 augmented
matrix whose fifth column is a pivot column. Is the system
consistent? Why (or why not)?
[2 3 h]
25. Suppose the coefficient matrix of a system of linear equations
has a pivot position in every row. Explain why the system is
1 -3
17.
18.
consistent.
In Exercises 19 and 20, choose h and k such that the system has (a)
no solution, (b) a unique solution, and (c) many solutions. Give
separate answers for each part.
26. Suppose the coefficient matrix of a linear system of three equa-
tions 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 so-
lution is unique if and only if.
19. x + hx2 = 2
20. xi + 3x2 2
4x1 + 8x2 = k
3x1 + hx2 = k
In Exercises 21 and 22, mark each statement True or False. Justify
each answer.
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 un-
knowns is sometimes called an underdetermined system. Sup-
pose that such a system happens to be consistent. Explain why
there must be an infinite number of solutions.
21. a. In some cases, a matrix may be row reduced to more than
one matrix in reduced echelon form, using different se-
quences of row operations.
b. The row reduction algorithm applies only to augmented
matrices for a linear system.
c. A basic variable in a linear system is a variable that corre-
sponds to a pivot column in the coefficient matrix.
30. Give an example of an inconsistent underdetermined system
of two equations in three unknowns.
31. A system of linear equations with more equations than un-
knowns is sometimes called an overdetermined system. Can
such a system be consistent? Illustrate your answer with a
specific system of three equations in two unknowns.
d. Finding a parametric description of the solution set of a
linear system is the same as solving the system.
e. If one row in an echelon form of an augmented matrix
is [0 0 0 5 0], then the associated linear system is
inconsistent.
32. Suppose an n x(n + 1) matrix is row reduced to reduced ech-
elon form. Approximately what fraction of the total number
of operations (flops) is involved in the backward phase of the
reduction when n = 30? when n = 300?
22. a. The echelon form of a matrix is unique.
b. The pivot positions in a matrix depend on whether row
interchanges are used in the row reduction process.
Suppose experimental data are represented by a set of points in
c. Reducing a matrix to echelon form is called the forward the plane. An interpolating polynomial for the data is a polyno-
mial 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.
phase of the row reduction process.
100% +
*True/false questions of this type will appear in many sections. Methods
for justifying your answers were described before Exercises 23 and 24 in
Section 1.1.
WEB
APR
1
1
étv
280
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