LINEAR ALGEBRA EXAM 1 T/F
QUESTIONS AND ANSWERS
Every elementary row operation is reversible (1.1) - Answer-True
A 5x6 matrix has 6 rows (1.1) - Answer-False - 5 rows and 6 columns
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. (1.1) - Answer-False - a solution set is
the set of all possible solutions of a linear system
Two fundamental questions about a linear system involve existence and uniqueness
(1.1) - Answer-True
Two matrices are row equivalent if they have the same number of rows (1.1) - Answer-
False - two matrices are row equivalent if there is a sequence of elementary row
operations that transforms one matrix into the other
Elementary row operations on an augmented matrix never change the solution set of
the associated linear system (1.1) - Answer-True
Two equivalent linear systems can have different solution sets (1.1) - Answer-False -
two linear systems are called equivalent if they have the same solution set
A consistent system of linear equations has one or more solutions (1.1) - Answer-False
- a system of linear equations is said to be consistent if it has either one solution or
infinitely many solutions
In some cases, a matrix may be row reduced to more than one matrix in reduced
echelon form, using different sequences of row operations (1.2) - Answer-False -
theorem 1 (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
(1.2) - Answer-False - the row reduction algorithm applies to any matrix whether or not
the matrix is viewed as an augmented matrix for a linear system. Any nonzero matrix
may be row reduced.
A basic variable in a linear system is a variable that corresponds to a pivot column in
the coefficient matrix (1.2) - Answer-True
Finding a parametric description of the solution of a linear system is the same as solving
the system (1.2) - Answer-True
, If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated
linear system is inconsistent (1.2) - Answer-False - the system would have infinitely
many solutions. An inconsistent system would have a row like [0 0 0 0 5]
The reduced echelon form of a matrix is unique (1.2) - Answer-True
If every column of an augmented matrix contains a pivot, then the corresponding
system is consistent (1.2) - Answer-False - the last column of an augmented matrix
should not have a pivot
The pivot positions in a matrix depend on whether row interchanges are used in the row
reduction process (1.2) - Answer-False - pivot positions are bounded to reduced
echelon form which is unique
A general solution of a system is an explicit description of all solutions of the system
(1.2) - Answer-True
Whenever a system has free variables, the solution set contains many solutions (1.2) -
Answer-False - it may contain a free variable but still be inconsistent
Another notation for the vector
[-4]
[ 3] is [-4 3] (1.3) - Answer-False - another notation would be (-4,3). [-4 3] would be a set
of two one-dimensional vectors
The points in the plane corresponding to
[-2]
[ 5] and
[-5]
[ 2] lie on a line through the origin (1.3) - Answer-False - if you were to graph the points
on a plane you would be able to see that no such line exists that would include both
points and the origin
An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1 (1.3) -
Answer-True
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 (1.3) - Answer-True
The set Span{u,v} is always visualized as a plane through the origin (1.3) - Answer-
False - if the vectors are linearly dependent, then their span will produce a line, not a
plane
When u and v are nonzero vectors, Span{u,v} contains only the line through u and the
origin, and the line through v and the origin (1.3) - Answer-False - Span{u,v} consists of
an entire plane that contains the origin when u and v are linearly independent
QUESTIONS AND ANSWERS
Every elementary row operation is reversible (1.1) - Answer-True
A 5x6 matrix has 6 rows (1.1) - Answer-False - 5 rows and 6 columns
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. (1.1) - Answer-False - a solution set is
the set of all possible solutions of a linear system
Two fundamental questions about a linear system involve existence and uniqueness
(1.1) - Answer-True
Two matrices are row equivalent if they have the same number of rows (1.1) - Answer-
False - two matrices are row equivalent if there is a sequence of elementary row
operations that transforms one matrix into the other
Elementary row operations on an augmented matrix never change the solution set of
the associated linear system (1.1) - Answer-True
Two equivalent linear systems can have different solution sets (1.1) - Answer-False -
two linear systems are called equivalent if they have the same solution set
A consistent system of linear equations has one or more solutions (1.1) - Answer-False
- a system of linear equations is said to be consistent if it has either one solution or
infinitely many solutions
In some cases, a matrix may be row reduced to more than one matrix in reduced
echelon form, using different sequences of row operations (1.2) - Answer-False -
theorem 1 (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
(1.2) - Answer-False - the row reduction algorithm applies to any matrix whether or not
the matrix is viewed as an augmented matrix for a linear system. Any nonzero matrix
may be row reduced.
A basic variable in a linear system is a variable that corresponds to a pivot column in
the coefficient matrix (1.2) - Answer-True
Finding a parametric description of the solution of a linear system is the same as solving
the system (1.2) - Answer-True
, If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated
linear system is inconsistent (1.2) - Answer-False - the system would have infinitely
many solutions. An inconsistent system would have a row like [0 0 0 0 5]
The reduced echelon form of a matrix is unique (1.2) - Answer-True
If every column of an augmented matrix contains a pivot, then the corresponding
system is consistent (1.2) - Answer-False - the last column of an augmented matrix
should not have a pivot
The pivot positions in a matrix depend on whether row interchanges are used in the row
reduction process (1.2) - Answer-False - pivot positions are bounded to reduced
echelon form which is unique
A general solution of a system is an explicit description of all solutions of the system
(1.2) - Answer-True
Whenever a system has free variables, the solution set contains many solutions (1.2) -
Answer-False - it may contain a free variable but still be inconsistent
Another notation for the vector
[-4]
[ 3] is [-4 3] (1.3) - Answer-False - another notation would be (-4,3). [-4 3] would be a set
of two one-dimensional vectors
The points in the plane corresponding to
[-2]
[ 5] and
[-5]
[ 2] lie on a line through the origin (1.3) - Answer-False - if you were to graph the points
on a plane you would be able to see that no such line exists that would include both
points and the origin
An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1 (1.3) -
Answer-True
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 (1.3) - Answer-True
The set Span{u,v} is always visualized as a plane through the origin (1.3) - Answer-
False - if the vectors are linearly dependent, then their span will produce a line, not a
plane
When u and v are nonzero vectors, Span{u,v} contains only the line through u and the
origin, and the line through v and the origin (1.3) - Answer-False - Span{u,v} consists of
an entire plane that contains the origin when u and v are linearly independent
