COMPUTATIONAL LINEAR ALGEBRA
EXAM 1 QUESTIONS WITH CORRECT
ANSWERS
Row Echelon Form (REF) - Answer-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 Row Echelon Form (RREF) - Answer-(In REF)
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.
4. All leading entries are 1
5. In a column with a leading 1, all other elements are 0.
Uniqueness of the Reduced Echelon Form - Answer-Each matrix is row equivalent to
one and only one reduced echelon matrix.
(Is it consistent? If so, is there only one solution(free variables or not)?)
pivot column - Answer-A column that contains a pivot position
pivot positon - Answer-In A is a location that corresponds to a pivot
Rank(A) (rank of A) - Answer-the number of pivot positions in a matrix A in ref and rref
(should be equal for ref and rref)
(True or False?): If the system is inconsistent, the solution set is empty, even when the
system has free variables. - Answer-True
What is the name given to the variables that correspond with the pivot columns? -
Answer-pivot variables or (basic variables, if you are a weirdo)
What is the name given to the variables that correspond with the non-pivot columns
within the matrix? - Answer-Free variables
When is a linear system considered "consistent?" - Answer-When the rightmost column
of the augmented matrix is not a pivot column.
If a linear system is consistent, what are the two possible outcomes associated with a
system considered as such? - Answer-1) It has a unique solution when there is no free
variable.
, 2) It has infinitely many solutions when there is at least one free variable.
Rouché-Capelli Theorem. (consistency of a linear system defined by ranks) - Answer-
The linear system Ax=b is consistent if and only if :
rank(A) = rank([ A b]) where A is m x n
According to the Rouché-Capelli Theorem, when is a linear system consistent, and how
do we differentiate between unique and infinite solutions. - Answer-Ax=b where A is size
mxn
Consistency: rank(A) = rank ([A b])
1) It has a unique solution when rank(A) = n.
2) It has infinitely many solutions when rank(A) < n.
If a matrix A is a certain size m x n, what does m represent, and what does n represent?
(We could just as easily say it is size X x Y or n x m?) - Answer-m = number of rows
n = number of columns
but really, the number of rows is the left integer m (or X or n)
and really number of columns is the right integer n (or Y or m)
If a matrix A is size m x n, and we wanted to multiply it times matrix b, what size would b
have to "be," no pun intended, in order for the multiplication to be defined? - Answer-
Matrix b would have to have n number of rows, but could have any number of columns,
so size n x w, where w is any integer.
What does the set of all scalar multiples of a nonzero vector U mean geometrically? {cU
| c ∈ R } - Answer-It is a line through the origin and U. (origin when c = 0, and then all
the other values of c are what causes the line to "form")
If b = c₁v₁ + c₂v₂ +c(i)v(i), {v(i) ∈ Rⁿ}, then what is this called, and what are the c(i)
known as? - Answer-Linear combination of v₁, v₂, ..., v(i), and the c(i)s are known as
weights.
True or False? A vector equation b has the same solution set as the linear system
Ax=b. - Answer-True
span{v₁, ..., v(p)} = - Answer-{c₁v₁ + ... + c(p)v(p)}, c(p) is are scalars
if S= span{v₁, ..., v₂}, then what does S contain? - Answer-S contains every scalar
multiple of a vector v(i),
where (i = 1:p)
Which statements are logically equivalent to:
1) A vector b is in span{a₁, ..., aₙ}? - Answer-2) A vector equation x₁a₁ +x₂a₂ + ... + xₙaₙ = b
has a solution (weights).
3) A linear system Ax=b with an augmented matrix
EXAM 1 QUESTIONS WITH CORRECT
ANSWERS
Row Echelon Form (REF) - Answer-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 Row Echelon Form (RREF) - Answer-(In REF)
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.
4. All leading entries are 1
5. In a column with a leading 1, all other elements are 0.
Uniqueness of the Reduced Echelon Form - Answer-Each matrix is row equivalent to
one and only one reduced echelon matrix.
(Is it consistent? If so, is there only one solution(free variables or not)?)
pivot column - Answer-A column that contains a pivot position
pivot positon - Answer-In A is a location that corresponds to a pivot
Rank(A) (rank of A) - Answer-the number of pivot positions in a matrix A in ref and rref
(should be equal for ref and rref)
(True or False?): If the system is inconsistent, the solution set is empty, even when the
system has free variables. - Answer-True
What is the name given to the variables that correspond with the pivot columns? -
Answer-pivot variables or (basic variables, if you are a weirdo)
What is the name given to the variables that correspond with the non-pivot columns
within the matrix? - Answer-Free variables
When is a linear system considered "consistent?" - Answer-When the rightmost column
of the augmented matrix is not a pivot column.
If a linear system is consistent, what are the two possible outcomes associated with a
system considered as such? - Answer-1) It has a unique solution when there is no free
variable.
, 2) It has infinitely many solutions when there is at least one free variable.
Rouché-Capelli Theorem. (consistency of a linear system defined by ranks) - Answer-
The linear system Ax=b is consistent if and only if :
rank(A) = rank([ A b]) where A is m x n
According to the Rouché-Capelli Theorem, when is a linear system consistent, and how
do we differentiate between unique and infinite solutions. - Answer-Ax=b where A is size
mxn
Consistency: rank(A) = rank ([A b])
1) It has a unique solution when rank(A) = n.
2) It has infinitely many solutions when rank(A) < n.
If a matrix A is a certain size m x n, what does m represent, and what does n represent?
(We could just as easily say it is size X x Y or n x m?) - Answer-m = number of rows
n = number of columns
but really, the number of rows is the left integer m (or X or n)
and really number of columns is the right integer n (or Y or m)
If a matrix A is size m x n, and we wanted to multiply it times matrix b, what size would b
have to "be," no pun intended, in order for the multiplication to be defined? - Answer-
Matrix b would have to have n number of rows, but could have any number of columns,
so size n x w, where w is any integer.
What does the set of all scalar multiples of a nonzero vector U mean geometrically? {cU
| c ∈ R } - Answer-It is a line through the origin and U. (origin when c = 0, and then all
the other values of c are what causes the line to "form")
If b = c₁v₁ + c₂v₂ +c(i)v(i), {v(i) ∈ Rⁿ}, then what is this called, and what are the c(i)
known as? - Answer-Linear combination of v₁, v₂, ..., v(i), and the c(i)s are known as
weights.
True or False? A vector equation b has the same solution set as the linear system
Ax=b. - Answer-True
span{v₁, ..., v(p)} = - Answer-{c₁v₁ + ... + c(p)v(p)}, c(p) is are scalars
if S= span{v₁, ..., v₂}, then what does S contain? - Answer-S contains every scalar
multiple of a vector v(i),
where (i = 1:p)
Which statements are logically equivalent to:
1) A vector b is in span{a₁, ..., aₙ}? - Answer-2) A vector equation x₁a₁ +x₂a₂ + ... + xₙaₙ = b
has a solution (weights).
3) A linear system Ax=b with an augmented matrix
