MATH 225 Final!!!MOSTLY TESTED - RATED 100%
CORRECT...GUARANTEED PASS
Augmented Matrix - (answers)a coefficient matrix with an extra column containing the
constant terms
Consistent system - (answers)a system of equations or inequalities that has at least one
solution
Is the statement "Every elementary row operation is reversible" true or false? -
(answers)True, because replacement, interchanging, and scaling are all reversible.
Is the statement "A
5×6
matrix has six rows" true or false? - (answers)False, because a
5×6
matrix has five rows and six columns.
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 subbed for x1...xn respectively" true or false? - (answers)False,
because the description applies to a single solution. The solution set consists of all possible
solutions.
Is the statement "Two fundamental questions about a linear system involve existence and
uniqueness" true or false? Explain. - (answers)True, because two fundamental questions
address whether the solution exists and whether there is only one solution.
Echelon Form - (answers)1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row
above it.
3. All entries in a column below a leading entry are zeros.
,Reduced Echelon Form - (answers)Same as echelon form, except all leading entries are 1;
each leading 1 is the only non-zero entry in its row; there is only one unique reduced
echelon form for every matrix
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon
form, using different sequences of row operations.
Is this statement true or false? - (answers)The statement is false. Each matrix is row
equivalent to one and only one reduced echelon matrix.
The row reduction algorithm applies only to augmented matrices for a linear system.
Is this statement true or false? - (answers)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.
A basic variable in a linear system is a variable that corresponds to a pivot column in the
coefficient matrix.
Is this statement true or false? - (answers)The statement is true. It is the definition of a basic
variable.
Finding a parametric description of the solution set of a linear system is the same as solving
the system.
Is this statement true or false? - (answers)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.
f one row in an echelon form of an augmented matrix is [0 0 0 5 0] then the associated linear
system is inconsistent.
Is this statement true or false? - (answers)The statement is false. The indicated row
corresponds to the equation
5x 4x4equals=0,
which does not by itself make the system inconsistent.
, What must be true of a linear system for it to have a unique solution? Select all that apply. -
(answers)The system is consistent, the system has no free variables
Use the augmented matrix to determine if the linear system is consistent. Is the linear
system represented by the augmented matrix consistent? - (answers)Yes, because the
rightmost column of the augmented matrix is not a pivot column.
HW 2 #7 - (answers)Rework
Existence and Uniqueness Theorem - (answers)A linear system is consistent if and only if the
rightmost column of the augmented matrix is not a pivot column. That is, if and only if an
echelon form of the augmented matrix has no row of the form [0...0 b] with b nonzero.
Suppose a
6*8 coefficient matrix for a system has
six pivot columns. Is the system consistent? Why or why not? - (answers)There is a pivot
position in each row of the coefficient matrix. The augmented matrix will have nine columns
and will not have a row of the form [0 0 0 0 0 0 0 0 1], so it is consistent
A system of linear equations with fewer equations than unknowns is sometimes called an
underdetermined system. Can such a system have a unique solution? Explain. - (answers)No,
it cannot have a unique solution. Because there are more variables than equations, there
must be at least one free variable. If the linear system is consistent and there is at least one
free variable, the solution set contains infinitely many solutions. If the linear system is
inconsistent, there is no solution.
a. When u and v are nonzero vectors, Span{u,v} contains only the line through u and the line
through v and the origin. - (answers)False. Span{u,v} includes linear combinations of both u
and v.
Any list of five real numbers is a vector in R5 - (answers)True. R5 denotes the collection of all
lists of five real numbers
CORRECT...GUARANTEED PASS
Augmented Matrix - (answers)a coefficient matrix with an extra column containing the
constant terms
Consistent system - (answers)a system of equations or inequalities that has at least one
solution
Is the statement "Every elementary row operation is reversible" true or false? -
(answers)True, because replacement, interchanging, and scaling are all reversible.
Is the statement "A
5×6
matrix has six rows" true or false? - (answers)False, because a
5×6
matrix has five rows and six columns.
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 subbed for x1...xn respectively" true or false? - (answers)False,
because the description applies to a single solution. The solution set consists of all possible
solutions.
Is the statement "Two fundamental questions about a linear system involve existence and
uniqueness" true or false? Explain. - (answers)True, because two fundamental questions
address whether the solution exists and whether there is only one solution.
Echelon Form - (answers)1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row
above it.
3. All entries in a column below a leading entry are zeros.
,Reduced Echelon Form - (answers)Same as echelon form, except all leading entries are 1;
each leading 1 is the only non-zero entry in its row; there is only one unique reduced
echelon form for every matrix
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon
form, using different sequences of row operations.
Is this statement true or false? - (answers)The statement is false. Each matrix is row
equivalent to one and only one reduced echelon matrix.
The row reduction algorithm applies only to augmented matrices for a linear system.
Is this statement true or false? - (answers)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.
A basic variable in a linear system is a variable that corresponds to a pivot column in the
coefficient matrix.
Is this statement true or false? - (answers)The statement is true. It is the definition of a basic
variable.
Finding a parametric description of the solution set of a linear system is the same as solving
the system.
Is this statement true or false? - (answers)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.
f one row in an echelon form of an augmented matrix is [0 0 0 5 0] then the associated linear
system is inconsistent.
Is this statement true or false? - (answers)The statement is false. The indicated row
corresponds to the equation
5x 4x4equals=0,
which does not by itself make the system inconsistent.
, What must be true of a linear system for it to have a unique solution? Select all that apply. -
(answers)The system is consistent, the system has no free variables
Use the augmented matrix to determine if the linear system is consistent. Is the linear
system represented by the augmented matrix consistent? - (answers)Yes, because the
rightmost column of the augmented matrix is not a pivot column.
HW 2 #7 - (answers)Rework
Existence and Uniqueness Theorem - (answers)A linear system is consistent if and only if the
rightmost column of the augmented matrix is not a pivot column. That is, if and only if an
echelon form of the augmented matrix has no row of the form [0...0 b] with b nonzero.
Suppose a
6*8 coefficient matrix for a system has
six pivot columns. Is the system consistent? Why or why not? - (answers)There is a pivot
position in each row of the coefficient matrix. The augmented matrix will have nine columns
and will not have a row of the form [0 0 0 0 0 0 0 0 1], so it is consistent
A system of linear equations with fewer equations than unknowns is sometimes called an
underdetermined system. Can such a system have a unique solution? Explain. - (answers)No,
it cannot have a unique solution. Because there are more variables than equations, there
must be at least one free variable. If the linear system is consistent and there is at least one
free variable, the solution set contains infinitely many solutions. If the linear system is
inconsistent, there is no solution.
a. When u and v are nonzero vectors, Span{u,v} contains only the line through u and the line
through v and the origin. - (answers)False. Span{u,v} includes linear combinations of both u
and v.
Any list of five real numbers is a vector in R5 - (answers)True. R5 denotes the collection of all
lists of five real numbers
