LINEAR ALGEBRA – QUIZ GUIDE
QUESTIONS WITH CORRECT ANSWERS
(1.1) Every elementary row operation is reversible. - Answer-True:
Replacement, interchanging, and scaling are all reversible.
(1.1) Elementary row operations on an augmented matrix never change the solution set
of the associated linear system. - Answer-True:
Each elementary row operation replaces a system with an equivalent system.
(1.1) A 5x6 matrix has 6 rows. - Answer-False:
5x6 matrix has 5 rows and 6 columns.
(1.1) Two matrices are row equivalent if they have the same number of rows. - Answer-
False:
Two matrices are row equivalent if there exists a sequence of elementary row
operations that transforms one matrix into the other.
(1.1) Two fundamental questions about a linear system involve existence and
uniqueness. - Answer-True:
The statement is true. The two fundamental questions are about whether the solution
exists and whether there is only one solution.
(1.1) Is matrix a, b and c in reduced echelon form, echelon form only or neither? -
Answer-a. reduced echelon form
b. echelon form
c. neither
(1.2)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-False:
Each matrix is row equivalent to one and only one reduced echelon matrix.
(1.2)The echelon form of a matrix is unique. - Answer-False:
The echelon form of a matrix is not unique, but the reduced echelon form is unique.
(1.2)The row reduction algorithm applies only to augmented matrices for a linear
system. - Answer-False:
The algorithm applies to any matrix, whether or not the matrix is viewed as an
augmented matrix for a linear system.
(1.2)The pivot positions in a matrix depend on whether row interchanges are used in the
row reduction process. - Answer-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.
, (1.2)Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - Answer-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.
(1.2)Suppose a 4x6 coefficient matrix for a system has four pivot columns. Is the system
consistent? Why or why not? - Answer-Consistent:
There is a pivot position in each row of the coefficient matrix. The augmented matrix will
have seven columns and will not have a row of the form
[0 0 0 0 0 0 1]
(1.2)Suppose the coefficient matrix of a system of linear equations has a pivot position
in every row. Explain why the system is consistent. - Answer-Consistent:
rightmost column of the augmented matrix is not a pivot column.
(1.3)Any list of five real numbers is a vector in R^5 - Answer-True:
R^5 denotes the collection of all lists of five real numbers.
(1.3)An example of a linear combination of vectors v₁ and v₂ term-20is the vector 0.5v₁ -
Answer-True:
0.5v₁ = 0.5v₁ + 0v₂
(1.3)The weights c₁,...,cᵣ in a linear combination c₁v₁+...+cᵣvᵣ cannot all be zero -
Answer-False:
Setting all the weights to zero results in the vector 0
(1.3)The solution set of the linear system whose augmented matrix is [a₁ a₂ a₃ b] is the
same as the solution set of the equation x₁a₁+x₂a₂+x₃a₃=b - Answer-True:
augmented matrix for x₁a₁+x₂a₂+x₃a₃=b is [a₁ a₂ a₃ b]
(1.3)The set Span {u, v} is always visualized as a plane through the origin. - Answer-
False:
It is often true, but
Span {u, v} is not a plane when v is a multiple of u or when u is the zero vector.
(1.3)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} - Answer-True:
The linear system corresponding to [a₁ a₂ a₃ b] has a solution when b can be written as
a linear combination of a₁,a₂ and a₃. This is equivalent to saying that span b is in the
Span {a₁,a₂,a₃}
(1.4) Every matrix equation Ax=b corresponds to a vector equation with the same
solution set. - Answer-True:
QUESTIONS WITH CORRECT ANSWERS
(1.1) Every elementary row operation is reversible. - Answer-True:
Replacement, interchanging, and scaling are all reversible.
(1.1) Elementary row operations on an augmented matrix never change the solution set
of the associated linear system. - Answer-True:
Each elementary row operation replaces a system with an equivalent system.
(1.1) A 5x6 matrix has 6 rows. - Answer-False:
5x6 matrix has 5 rows and 6 columns.
(1.1) Two matrices are row equivalent if they have the same number of rows. - Answer-
False:
Two matrices are row equivalent if there exists a sequence of elementary row
operations that transforms one matrix into the other.
(1.1) Two fundamental questions about a linear system involve existence and
uniqueness. - Answer-True:
The statement is true. The two fundamental questions are about whether the solution
exists and whether there is only one solution.
(1.1) Is matrix a, b and c in reduced echelon form, echelon form only or neither? -
Answer-a. reduced echelon form
b. echelon form
c. neither
(1.2)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-False:
Each matrix is row equivalent to one and only one reduced echelon matrix.
(1.2)The echelon form of a matrix is unique. - Answer-False:
The echelon form of a matrix is not unique, but the reduced echelon form is unique.
(1.2)The row reduction algorithm applies only to augmented matrices for a linear
system. - Answer-False:
The algorithm applies to any matrix, whether or not the matrix is viewed as an
augmented matrix for a linear system.
(1.2)The pivot positions in a matrix depend on whether row interchanges are used in the
row reduction process. - Answer-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.
, (1.2)Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - Answer-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.
(1.2)Suppose a 4x6 coefficient matrix for a system has four pivot columns. Is the system
consistent? Why or why not? - Answer-Consistent:
There is a pivot position in each row of the coefficient matrix. The augmented matrix will
have seven columns and will not have a row of the form
[0 0 0 0 0 0 1]
(1.2)Suppose the coefficient matrix of a system of linear equations has a pivot position
in every row. Explain why the system is consistent. - Answer-Consistent:
rightmost column of the augmented matrix is not a pivot column.
(1.3)Any list of five real numbers is a vector in R^5 - Answer-True:
R^5 denotes the collection of all lists of five real numbers.
(1.3)An example of a linear combination of vectors v₁ and v₂ term-20is the vector 0.5v₁ -
Answer-True:
0.5v₁ = 0.5v₁ + 0v₂
(1.3)The weights c₁,...,cᵣ in a linear combination c₁v₁+...+cᵣvᵣ cannot all be zero -
Answer-False:
Setting all the weights to zero results in the vector 0
(1.3)The solution set of the linear system whose augmented matrix is [a₁ a₂ a₃ b] is the
same as the solution set of the equation x₁a₁+x₂a₂+x₃a₃=b - Answer-True:
augmented matrix for x₁a₁+x₂a₂+x₃a₃=b is [a₁ a₂ a₃ b]
(1.3)The set Span {u, v} is always visualized as a plane through the origin. - Answer-
False:
It is often true, but
Span {u, v} is not a plane when v is a multiple of u or when u is the zero vector.
(1.3)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} - Answer-True:
The linear system corresponding to [a₁ a₂ a₃ b] has a solution when b can be written as
a linear combination of a₁,a₂ and a₃. This is equivalent to saying that span b is in the
Span {a₁,a₂,a₃}
(1.4) Every matrix equation Ax=b corresponds to a vector equation with the same
solution set. - Answer-True:
