Linear Algebra
1.1
10. a.
Is the statement "Every elementary row operation is reversible" true or
false? Explain. - ANSWER>>>True, because replacement, interchanging,
and scaling are all reversible.
1.1
10. b.
Is the statement A 5×6 matrix has six rows" true or false? Explain. -
ANSWER>>>False, because a 5×6 matrix has five rows and six columns.
1.1
10. c.
Is the statement "The solution set of a linear system involving variables x1,
..., xn is a list of numbers (s1, ..., sn) that makes each equation in the
system a true statement when the values s1, ..., sn are substituted for x1,
..., xn, respectively" true or false? Explain. - ANSWER>>>False, because
the description applies to a single solution. The solution set consists of all
possible solutions.
1.1
10. d.
Is the statement "Two fundamental questions about a linear system involve
existence and uniqueness" true or false? Explain. - ANSWER>>>True,
because two fundamental questions address whether the solution exists
and whether there is only one solution.
1.1
11. a.
Is the statement "Two matrices are row equivalent if they have the same
number of rows" true or false? Explain. - ANSWER>>>False, because if
two matrices are row equivalent it means that there exists a sequence of
row operations that transforms one matrix to the other.
,1.1
11. b.
Is the statement "Elementary row operations on an augmented matrix
never change the solution set of the associated linear system" true or
false? Explain. - ANSWER>>>True, because the elementary row
operations replace a system with an equivalent system.
1.1
11. c.
Is the statement "Two equivalent linear systems can have different solution
sets" true or false? Explain. - ANSWER>>>False, because two systems
are called equivalent if they have the same solution set.
1.1
11. d.
Is the statement "A consistent system of linear equations has one or more
solutions" true or false? Explain. - ANSWER>>>True, a consistent system
is defined as a system that has at least one solution.
1.2
6. a.
In some cases, a matrix may be row reduced to more than one matrix in
reduced echelon form, using different sequences of row operations. -
ANSWER>>>The statement is false. Each matrix is row equivalent to one
and only one reduced echelon matrix.
1.2
6. b.
The row reduction algorithm applies only to augmented matrices for a linear
system. - ANSWER>>>The statement is false. The algorithm applies to any
matrix, whether or not the matrix is viewed as an augmented matrix for a
linear system.
1.2
,6. c.
A basic variable in a linear system is a variable that corresponds to a pivot
column in the coefficient matrix. - ANSWER>>>The statement is true. It is
the definition of a basic variable.
1.2
6. d.
Finding a parametric description of the solution set of a linear system is the
same as solving the system. - ANSWER>>>The statement is false. The
solution set of a linear system can only be expressed using a parametric
description if the system has at least one solution.
1.2
6. e.
If one row in an echelon form of an augmented matrix is left bracket
[0 0 0 5 0 ]
then the associated linear system is inconsistent. - ANSWER>>>The
statement is false. The indicated row corresponds to the equation
5x(subscript 4) = 0,
which does not by itself make the system inconsistent.
1.2
7. Part 1
Suppose the coefficient matrix of a linear system of four equations in four
variables has a pivot in each column. Explain why the system has a unique
solution.
What must be true of a linear system for it to have a unique solution?
Select all that apply. - ANSWER>>>The system is consistent.
The system has no free variables.
1.2
7. Part 2
Use the given assumption that the coefficient matrix of the linear system of
four equations in four variables has a pivot in each column to determine the
, dimensions of the coefficient matrix. - ANSWER>>>The coefficient matrix
has
four rows and four columns.
1.2
7. Part 3
Let the coefficient matrix be in reduced echelon form with a pivot in each
column, since each matrix is equivalent to one and only one reduced
echelon matrix. Construct a matrix with the dimensions determined in the
previous step that is in reduced echelon form and has a pivot in each
column. - ANSWER>>>[ 1 0 0 0 ]
[0100]
[0010]
[0001]
1.2
7. Part 4
Now find an augmented matrix in reduced echelon form that represents a
linear system of four equations in four variables for which the
corresponding coefficient matrix has a pivot in each column. Choose the
correct ANSWER below. - ANSWER>>>[ 1 0 0 0 a ]
[0100b]
[0010c]
[0001d]
1.2
7. Part 5
Use the augmented matrix to determine if the linear system is consistent. Is
the linear system represented by the augmented matrix consistent? -
ANSWER>>>Yes, because the rightmost column of the augmented matrix
is not a pivot column.
1.2
7. Part 6
1.1
10. a.
Is the statement "Every elementary row operation is reversible" true or
false? Explain. - ANSWER>>>True, because replacement, interchanging,
and scaling are all reversible.
1.1
10. b.
Is the statement A 5×6 matrix has six rows" true or false? Explain. -
ANSWER>>>False, because a 5×6 matrix has five rows and six columns.
1.1
10. c.
Is the statement "The solution set of a linear system involving variables x1,
..., xn is a list of numbers (s1, ..., sn) that makes each equation in the
system a true statement when the values s1, ..., sn are substituted for x1,
..., xn, respectively" true or false? Explain. - ANSWER>>>False, because
the description applies to a single solution. The solution set consists of all
possible solutions.
1.1
10. d.
Is the statement "Two fundamental questions about a linear system involve
existence and uniqueness" true or false? Explain. - ANSWER>>>True,
because two fundamental questions address whether the solution exists
and whether there is only one solution.
1.1
11. a.
Is the statement "Two matrices are row equivalent if they have the same
number of rows" true or false? Explain. - ANSWER>>>False, because if
two matrices are row equivalent it means that there exists a sequence of
row operations that transforms one matrix to the other.
,1.1
11. b.
Is the statement "Elementary row operations on an augmented matrix
never change the solution set of the associated linear system" true or
false? Explain. - ANSWER>>>True, because the elementary row
operations replace a system with an equivalent system.
1.1
11. c.
Is the statement "Two equivalent linear systems can have different solution
sets" true or false? Explain. - ANSWER>>>False, because two systems
are called equivalent if they have the same solution set.
1.1
11. d.
Is the statement "A consistent system of linear equations has one or more
solutions" true or false? Explain. - ANSWER>>>True, a consistent system
is defined as a system that has at least one solution.
1.2
6. a.
In some cases, a matrix may be row reduced to more than one matrix in
reduced echelon form, using different sequences of row operations. -
ANSWER>>>The statement is false. Each matrix is row equivalent to one
and only one reduced echelon matrix.
1.2
6. b.
The row reduction algorithm applies only to augmented matrices for a linear
system. - ANSWER>>>The statement is false. The algorithm applies to any
matrix, whether or not the matrix is viewed as an augmented matrix for a
linear system.
1.2
,6. c.
A basic variable in a linear system is a variable that corresponds to a pivot
column in the coefficient matrix. - ANSWER>>>The statement is true. It is
the definition of a basic variable.
1.2
6. d.
Finding a parametric description of the solution set of a linear system is the
same as solving the system. - ANSWER>>>The statement is false. The
solution set of a linear system can only be expressed using a parametric
description if the system has at least one solution.
1.2
6. e.
If one row in an echelon form of an augmented matrix is left bracket
[0 0 0 5 0 ]
then the associated linear system is inconsistent. - ANSWER>>>The
statement is false. The indicated row corresponds to the equation
5x(subscript 4) = 0,
which does not by itself make the system inconsistent.
1.2
7. Part 1
Suppose the coefficient matrix of a linear system of four equations in four
variables has a pivot in each column. Explain why the system has a unique
solution.
What must be true of a linear system for it to have a unique solution?
Select all that apply. - ANSWER>>>The system is consistent.
The system has no free variables.
1.2
7. Part 2
Use the given assumption that the coefficient matrix of the linear system of
four equations in four variables has a pivot in each column to determine the
, dimensions of the coefficient matrix. - ANSWER>>>The coefficient matrix
has
four rows and four columns.
1.2
7. Part 3
Let the coefficient matrix be in reduced echelon form with a pivot in each
column, since each matrix is equivalent to one and only one reduced
echelon matrix. Construct a matrix with the dimensions determined in the
previous step that is in reduced echelon form and has a pivot in each
column. - ANSWER>>>[ 1 0 0 0 ]
[0100]
[0010]
[0001]
1.2
7. Part 4
Now find an augmented matrix in reduced echelon form that represents a
linear system of four equations in four variables for which the
corresponding coefficient matrix has a pivot in each column. Choose the
correct ANSWER below. - ANSWER>>>[ 1 0 0 0 a ]
[0100b]
[0010c]
[0001d]
1.2
7. Part 5
Use the augmented matrix to determine if the linear system is consistent. Is
the linear system represented by the augmented matrix consistent? -
ANSWER>>>Yes, because the rightmost column of the augmented matrix
is not a pivot column.
1.2
7. Part 6
