LINEAR ALGEBRA EXAM 1 QUESTIONS
AND ANSWERS
True, because replacement, interchanging, and scaling are all reversible. - Answer-Is
the statement "Every elementary row operation is reversible" true or false?
False, because a 5x6 matrix has five rows and six columns. - Answer-Id the statement
"A 5x6 matrix has six rows" true or false?
False, because the description applies to a single solution. The solution set consists of
all possible solutions. - Answer-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?
True - Answer-Is the statement "Two fundamental questions about a linear system
involve existence and uniqueness" true or false?
False, each matrix is row equivalent to one and only one reduced echelon matrix. -
Answer-In some cases, a matrix may be row reduced to more than echelon form, using
different sequences of row operations. True or false?
False, the algorithm applies to any matrix, whether or not the matrix is viewed as an
augmented matrix for a linear system. - Answer-The row reduction algorithm applies
only to augmented matrices for a linear system. True or false?
True, It is the definition of a basic variable. - Answer-A basic variable in a linear system
is a variable that corresponds to a pivot column in the coefficient matrix. True or false.
False, the solution set of a linear system can only be expressed using a parametric
description if the system has at lease one solution. - Answer-Finding a parametric
description of the solution set of a linear system is the same as solving the system. True
or False?
False, the indicated row corresponds to the equation 5x4=0, which does not by itself
make the system inconsistent. - Answer-If one row in echelon form of an augmented
matrix is [0 0 0 5 0], then the associated linear system is inconsistent. True or false?
False, the echelon form of a matrix is always unique, but the reduced echelon form is
unique. - Answer-The echelon form of a matrix is unique. True or false?
False. The pivot position in a matrix are determined completely by the positions of the
leading entries in the nonzero rows of any echelon form obtained from the matrix. -
, Answer-The pivot positions in a matrix depend on whether row interchanges are used in
the row reduction process. True or false?
True. Reducing a matrix to echelon form is called the forward phase and reducing a
matrix to reduced echelon form is called the backward phase. - Answer-Reducing a
matrix to echelon form is called the forward phase of the row reduction process. True or
false?
False. The existence of at least one solution is not related to the presence or absence
of free variables. If the system is inconsistent, the solution set is empty. - Answer-
Whenever a system has a free variable, the solution set contains many solutions. True
or false?
True. The row reduction algorithm leads directly to an explicit description of the solution
set of a linear system when the algorithm is applied to the augmented matrix of the
system, leading to a general solution of a system. - Answer-A general solution of a
system is an explicit description of all solutions of the system. True or false?
Is not
No row - Answer-A linear system is consistent if and only if the rightmost column of the
augmented matrix ___________ a pivot column. That is, if and only if an echelon form
of the augmented matrix has
_________ of the form [0 ... 0 b] with b nonzero.
False. Span{u,v} includes linear combinations of both u and v. - Answer-When u and v
are nonzero vectors, Span{u,v} contains only the line through u and the line through v
and the origin. True or false?
True. ℝ^5 denotes the collection of all lists of five real numbers. - Answer-Any list of five
real numbers is a vector in ℝ^5. True or false?
True. An augmented matrix has a solution when the last column can be written as a
linear combination of the other columns. A linear system augmented has a solution
when the last column of its augmented matrix can be written as a linear combination of
the other columns. - Answer-Asking whether the linear system corresponding to an
augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span
{a1,a2,a3}. True or false?
False. Adding u−v to v results in u. - Answer-The vector v results when a vector u−v is
added to the vector v. True or false?
False. Setting all the weights equal to zero results in the vector 0. - Answer-The weights
c1,...,cp in a linear combination c1v1+..+cpvp cannot all be zeros. True or false?
False. The equation Ax=b is referred to as a matrix equation because A is a matrix. -
Answer-The equation Ax=b is referred to as a vector equation. True or false?
AND ANSWERS
True, because replacement, interchanging, and scaling are all reversible. - Answer-Is
the statement "Every elementary row operation is reversible" true or false?
False, because a 5x6 matrix has five rows and six columns. - Answer-Id the statement
"A 5x6 matrix has six rows" true or false?
False, because the description applies to a single solution. The solution set consists of
all possible solutions. - Answer-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?
True - Answer-Is the statement "Two fundamental questions about a linear system
involve existence and uniqueness" true or false?
False, each matrix is row equivalent to one and only one reduced echelon matrix. -
Answer-In some cases, a matrix may be row reduced to more than echelon form, using
different sequences of row operations. True or false?
False, the algorithm applies to any matrix, whether or not the matrix is viewed as an
augmented matrix for a linear system. - Answer-The row reduction algorithm applies
only to augmented matrices for a linear system. True or false?
True, It is the definition of a basic variable. - Answer-A basic variable in a linear system
is a variable that corresponds to a pivot column in the coefficient matrix. True or false.
False, the solution set of a linear system can only be expressed using a parametric
description if the system has at lease one solution. - Answer-Finding a parametric
description of the solution set of a linear system is the same as solving the system. True
or False?
False, the indicated row corresponds to the equation 5x4=0, which does not by itself
make the system inconsistent. - Answer-If one row in echelon form of an augmented
matrix is [0 0 0 5 0], then the associated linear system is inconsistent. True or false?
False, the echelon form of a matrix is always unique, but the reduced echelon form is
unique. - Answer-The echelon form of a matrix is unique. True or false?
False. The pivot position in a matrix are determined completely by the positions of the
leading entries in the nonzero rows of any echelon form obtained from the matrix. -
, Answer-The pivot positions in a matrix depend on whether row interchanges are used in
the row reduction process. True or false?
True. Reducing a matrix to echelon form is called the forward phase and reducing a
matrix to reduced echelon form is called the backward phase. - Answer-Reducing a
matrix to echelon form is called the forward phase of the row reduction process. True or
false?
False. The existence of at least one solution is not related to the presence or absence
of free variables. If the system is inconsistent, the solution set is empty. - Answer-
Whenever a system has a free variable, the solution set contains many solutions. True
or false?
True. The row reduction algorithm leads directly to an explicit description of the solution
set of a linear system when the algorithm is applied to the augmented matrix of the
system, leading to a general solution of a system. - Answer-A general solution of a
system is an explicit description of all solutions of the system. True or false?
Is not
No row - Answer-A linear system is consistent if and only if the rightmost column of the
augmented matrix ___________ a pivot column. That is, if and only if an echelon form
of the augmented matrix has
_________ of the form [0 ... 0 b] with b nonzero.
False. Span{u,v} includes linear combinations of both u and v. - Answer-When u and v
are nonzero vectors, Span{u,v} contains only the line through u and the line through v
and the origin. True or false?
True. ℝ^5 denotes the collection of all lists of five real numbers. - Answer-Any list of five
real numbers is a vector in ℝ^5. True or false?
True. An augmented matrix has a solution when the last column can be written as a
linear combination of the other columns. A linear system augmented has a solution
when the last column of its augmented matrix can be written as a linear combination of
the other columns. - Answer-Asking whether the linear system corresponding to an
augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span
{a1,a2,a3}. True or false?
False. Adding u−v to v results in u. - Answer-The vector v results when a vector u−v is
added to the vector v. True or false?
False. Setting all the weights equal to zero results in the vector 0. - Answer-The weights
c1,...,cp in a linear combination c1v1+..+cpvp cannot all be zeros. True or false?
False. The equation Ax=b is referred to as a matrix equation because A is a matrix. -
Answer-The equation Ax=b is referred to as a vector equation. True or false?
