LINEAR ALGEBRA EXAM STUDY GUIDE
SET QUESTIONS AND ANSWERS
Determine whether the statement below is true or false. Justify the answer.
Every elementary row operation is reversible. - Answer-The statement is true.
Replacement, interchanging, and scaling are all reversible.
Determine whether the statement below is true or false. Justify the answer.
Elementary row operations on an augmented matrix never change the solution set of
the associated linear system. - Answer-The statement is true. Each elementary row
operation replaces a system with an equivalent system.
Determine whether the statement below is true or false. Justify the answer.
A 5×6 matrix has six rows. - Answer-The statement is false. A
5×6
matrix has five rows and six columns.
Determine whether the statement below is true or false. Justify the answer.
Two matrices are row equivalent if they have the same number of rows. - Answer-The
statement is false. Two matrices are row equivalent if there exists a sequence of
elementary row operations that transforms one matrix into the other.
Determine whether the statement below is true or false. Justify the answer.
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. - Answer-The statement is false. The
given description is of a single solution of such a system. The solution set of the system
consists of all possible solutions.
Determine whether the statement below is true or false. Justify the answer.
Two fundamental questions about a linear system involve existence and uniqueness. -
Answer-The statement is true. The two fundamental questions are about whether the
solution exists and whether there is only one solution.
Determine whether the statement below is true or false. Justify the answer.
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.
Determine whether the statement below is true or false. Justify the answer.
The echelon form of a matrix is unique. - Answer-The statement is false. The echelon
form of a matrix is not unique, but the reduced echelon form is unique.
Determine whether the statement below is true or false. Justify the answer.
, 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.
Determine whether the statement below is true or false. Justify the answer.
The pivot positions in a matrix depend on whether row interchanges are used in the row
reduction process. - Answer-The statement is false. The pivot positions 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.
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.
Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - Answer-The statement is 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.
Suppose a 4×7 coefficient matrix for a system has four pivot columns. Is the system
consistent? Why or why not? - Answer-There is a pivot position in each row of the
coefficient matrix. The augmented matrix will have
eight
columns and will not have a row of the form
00000001
,
so the system is consistent.
Another notation for the vector
[−4
3]
is
[−4 3]
. - Answer-The statement is false. The alternative notation for a (column) vector is
(−4,3),
using parentheses and a comma.
Any list of five real numbers is a vector in
ℝ5. - Answer-The statement is true.
ℝ5
denotes the collection of all lists of five real numbers.
An example of a linear combination of vectors
v1
and
SET QUESTIONS AND ANSWERS
Determine whether the statement below is true or false. Justify the answer.
Every elementary row operation is reversible. - Answer-The statement is true.
Replacement, interchanging, and scaling are all reversible.
Determine whether the statement below is true or false. Justify the answer.
Elementary row operations on an augmented matrix never change the solution set of
the associated linear system. - Answer-The statement is true. Each elementary row
operation replaces a system with an equivalent system.
Determine whether the statement below is true or false. Justify the answer.
A 5×6 matrix has six rows. - Answer-The statement is false. A
5×6
matrix has five rows and six columns.
Determine whether the statement below is true or false. Justify the answer.
Two matrices are row equivalent if they have the same number of rows. - Answer-The
statement is false. Two matrices are row equivalent if there exists a sequence of
elementary row operations that transforms one matrix into the other.
Determine whether the statement below is true or false. Justify the answer.
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. - Answer-The statement is false. The
given description is of a single solution of such a system. The solution set of the system
consists of all possible solutions.
Determine whether the statement below is true or false. Justify the answer.
Two fundamental questions about a linear system involve existence and uniqueness. -
Answer-The statement is true. The two fundamental questions are about whether the
solution exists and whether there is only one solution.
Determine whether the statement below is true or false. Justify the answer.
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.
Determine whether the statement below is true or false. Justify the answer.
The echelon form of a matrix is unique. - Answer-The statement is false. The echelon
form of a matrix is not unique, but the reduced echelon form is unique.
Determine whether the statement below is true or false. Justify the answer.
, 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.
Determine whether the statement below is true or false. Justify the answer.
The pivot positions in a matrix depend on whether row interchanges are used in the row
reduction process. - Answer-The statement is false. The pivot positions 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.
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.
Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - Answer-The statement is 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.
Suppose a 4×7 coefficient matrix for a system has four pivot columns. Is the system
consistent? Why or why not? - Answer-There is a pivot position in each row of the
coefficient matrix. The augmented matrix will have
eight
columns and will not have a row of the form
00000001
,
so the system is consistent.
Another notation for the vector
[−4
3]
is
[−4 3]
. - Answer-The statement is false. The alternative notation for a (column) vector is
(−4,3),
using parentheses and a comma.
Any list of five real numbers is a vector in
ℝ5. - Answer-The statement is true.
ℝ5
denotes the collection of all lists of five real numbers.
An example of a linear combination of vectors
v1
and
