LINEAR ALGEBRA EXAM FINAL
QUESTIONS WITH CORRECT ANSWERS
a system of linear equations is said to be _____ if it has either one solution or infinitely
many solutions - Answer-consistent
a system of linear equations is said to be _____ if it has no solution - Answer-
inconsistent
elementary row operations - Answer-replacement (replace one row by itself and a
multiple of another row), interchange (interchange 2 rows), scaling (multiply all entries in
a row by a nonzero constant)
if the augmented matrices of two linear systems are row equivalent, then ... - Answer-
the two systems have the same solution set
augmented matrix - Answer-put the constants on the right-hand side next to the
coefficient matrix
A rectangular matrix is in echelon form (or row echelon form) if it has the 3 following
properties: - 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.
If a matrix in echelon form satisfies the following additional conditions, then it is in
reduced echelon form: - Answer-4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
Theorem: Uniqueness of the Reduced Echelon Form - Answer-each matrix is row
equivalent to one and only one reduced echelon matrix
a _______ in a matrix A is a location in A that corresponds to a leading 1 in the reduced
echelon form of A. A ______ is a column of A that contains a pivot position. - Answer-
pivot position, pivot column
Theorem: Existence and Uniqueness Theorem - Answer-A linear system is consistent if
and only if the rightmost column of the augmented matrix is not a pivot column-- that is,
if and only if an echelon form of the augmented matrix has no row of the form [ 0 ... 0 b]
with b nonzero.
If a linear system is consistent, then the solution set contains either (i) a unique solution,
when there are no free variables, or (ii) infinitely many solutions, when there is at least
one free variable.
,If a linear system is consistent, then the solution is unique if and only if... - Answer-every
column of the coefficient matrix is a pivot column; otherwise, there are infinitely many
solutions.
A matrix with only one column is called a - Answer-column vector
Parallelogram Rule for Addition - Answer-If u and v in R^2 are represented as points in
the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other
vertices are u, 0, and v.
The vector whose entries are all zero is called the - Answer-zero vector and is denoted
by 0.
A vector equation x1a1 + x2a2 +...+ xnan = b has the same solution set as the linear
system whose augmented matrix is ... - Answer-[ a1 a2 ... an b]
If v1, ..., vp are in R^n, then the set of all linear combinations of v1, ..., vp is denoted by
Span{v1, ..., vp} and is called the - Answer-subset of R^n spanned (or generated) by v1,
..., vp. That is Span{v1, ..., vp} is the collection of all vectors that can be written in the
form c1v1 + c2v2 + ... + cpvp with c1,..., cp scalars.
If A is an m x n matrix, with columns a1, ..., an, and if x is in R^n, then the product of A
and x, denoted by Ax, ... - Answer-is the linear combination of the columns of A using
the corresponding entries in x as weights
= x1a1 + x2a2 + ... + xnan
If A is an m x n matrix, with columns a1, ..., an, and if b is in R^m, the matrix equation
Ax = b has the same solution set as the vector equation x1a1 + x2a2 + ... + xnan = b,
which, in turn, has the same solution set as the system of linear equations whose
augmented matrix is - Answer-[ a1 a2 ... an b]
The equation Ax = b has a solution if and only if - Answer-b is a linear combination of
the columns of A.
Theorem: Let A be an m x n matrix. Then the following statements are logically
equivalent. That is, for a particular A either they are all true statements or they are all
false. (Warning: this is about a coefficient matrix, not an augmented matrix. If an
augmented matrix [A b] has a pivot position in every row, then the equation Ax = b may
or may not be consistent. - Answer-a. For each b in R^m, the equation Ax = b has a
solution.
b. Each b in R^m is a linear combination of the columns of A.
c. The columns of A span R^m.
d. A has a pivot position in every row.
, A system of linear equations is said to be _____ if it can be written in the form Ax = 0,
where A is an m x n matrix and 0 is the zero vector in R^m. This system always has at
least one solution, namely x = 0 (the zero vector). - Answer-homogeneous
the zero solution is usually called the ______ solution - Answer-trivial
_____ is a nonzero vector x that satisfies Ax= 0 - Answer-nontrivial solution
The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation
has ... - Answer-at least one free variable.
Theorem: Supposed the equation Ax=b is consistent for some given b, and let p be a
solution. Then the solutions set of Ax=b is the set of all vectors of the form w = p + vh,
where vh is any solution of the... Warning: only applies to an equation Ax = b that has at
least one nonzero solution p. When Ax= b has no solution, the solution set is empty. -
Answer-homogeneous equation Ax = 0.
An indexed set of vectors {v1, ..., vp} in R^n is said to be _______ if the vector equation
x1v1 + x2v2 + ... + xpvp = 0 has only the trivial solution. - Answer-linearly independent
The set of vectors {v1, ..., vp} is said to be _______ if there exists weights c1, ..., cp, not
all zero, such that c1v1 + c2v2 + ... + cpvp = 0 - Answer-linearly dependent
The columns of a matrix A are linearly independent if and only if the equation Ax= 0 has
only the _____ - Answer-trivial solution
A set of two vectors {v1, v2} is ________ if at least one of the vectors is a multiple of the
other. - Answer-linearly dependent
A set of two vectors {v1, v2} is ________ if and only if neither of the vectors is a multiple
of the other. - Answer-linearly independent
Theorem: If a set contains more vectors than there are entries in each vector, then the
set is _______. That is, any set {v1, ..., vp} in R^n is _______ if p > n. - Answer-linearly
dependent
Theorem: If a set S = {v1, ..., vp} in R^n contains the zero vector, then the set is
_______. - Answer-linearly dependent
A _______ (or function or mapping) T from R^n to R^m is a rule that assigns to each
vector x in R^n a vector T(x) in R^m. The set R^n is called the ____ of T, and R^m is
called the ____ of T. - Answer-transformation, domain, codomain
The transformation T: R^2 to R^2 defined by T(x) = Ax is called a ______ transformation
- Answer-shear
QUESTIONS WITH CORRECT ANSWERS
a system of linear equations is said to be _____ if it has either one solution or infinitely
many solutions - Answer-consistent
a system of linear equations is said to be _____ if it has no solution - Answer-
inconsistent
elementary row operations - Answer-replacement (replace one row by itself and a
multiple of another row), interchange (interchange 2 rows), scaling (multiply all entries in
a row by a nonzero constant)
if the augmented matrices of two linear systems are row equivalent, then ... - Answer-
the two systems have the same solution set
augmented matrix - Answer-put the constants on the right-hand side next to the
coefficient matrix
A rectangular matrix is in echelon form (or row echelon form) if it has the 3 following
properties: - 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.
If a matrix in echelon form satisfies the following additional conditions, then it is in
reduced echelon form: - Answer-4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
Theorem: Uniqueness of the Reduced Echelon Form - Answer-each matrix is row
equivalent to one and only one reduced echelon matrix
a _______ in a matrix A is a location in A that corresponds to a leading 1 in the reduced
echelon form of A. A ______ is a column of A that contains a pivot position. - Answer-
pivot position, pivot column
Theorem: Existence and Uniqueness Theorem - Answer-A linear system is consistent if
and only if the rightmost column of the augmented matrix is not a pivot column-- that is,
if and only if an echelon form of the augmented matrix has no row of the form [ 0 ... 0 b]
with b nonzero.
If a linear system is consistent, then the solution set contains either (i) a unique solution,
when there are no free variables, or (ii) infinitely many solutions, when there is at least
one free variable.
,If a linear system is consistent, then the solution is unique if and only if... - Answer-every
column of the coefficient matrix is a pivot column; otherwise, there are infinitely many
solutions.
A matrix with only one column is called a - Answer-column vector
Parallelogram Rule for Addition - Answer-If u and v in R^2 are represented as points in
the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other
vertices are u, 0, and v.
The vector whose entries are all zero is called the - Answer-zero vector and is denoted
by 0.
A vector equation x1a1 + x2a2 +...+ xnan = b has the same solution set as the linear
system whose augmented matrix is ... - Answer-[ a1 a2 ... an b]
If v1, ..., vp are in R^n, then the set of all linear combinations of v1, ..., vp is denoted by
Span{v1, ..., vp} and is called the - Answer-subset of R^n spanned (or generated) by v1,
..., vp. That is Span{v1, ..., vp} is the collection of all vectors that can be written in the
form c1v1 + c2v2 + ... + cpvp with c1,..., cp scalars.
If A is an m x n matrix, with columns a1, ..., an, and if x is in R^n, then the product of A
and x, denoted by Ax, ... - Answer-is the linear combination of the columns of A using
the corresponding entries in x as weights
= x1a1 + x2a2 + ... + xnan
If A is an m x n matrix, with columns a1, ..., an, and if b is in R^m, the matrix equation
Ax = b has the same solution set as the vector equation x1a1 + x2a2 + ... + xnan = b,
which, in turn, has the same solution set as the system of linear equations whose
augmented matrix is - Answer-[ a1 a2 ... an b]
The equation Ax = b has a solution if and only if - Answer-b is a linear combination of
the columns of A.
Theorem: Let A be an m x n matrix. Then the following statements are logically
equivalent. That is, for a particular A either they are all true statements or they are all
false. (Warning: this is about a coefficient matrix, not an augmented matrix. If an
augmented matrix [A b] has a pivot position in every row, then the equation Ax = b may
or may not be consistent. - Answer-a. For each b in R^m, the equation Ax = b has a
solution.
b. Each b in R^m is a linear combination of the columns of A.
c. The columns of A span R^m.
d. A has a pivot position in every row.
, A system of linear equations is said to be _____ if it can be written in the form Ax = 0,
where A is an m x n matrix and 0 is the zero vector in R^m. This system always has at
least one solution, namely x = 0 (the zero vector). - Answer-homogeneous
the zero solution is usually called the ______ solution - Answer-trivial
_____ is a nonzero vector x that satisfies Ax= 0 - Answer-nontrivial solution
The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation
has ... - Answer-at least one free variable.
Theorem: Supposed the equation Ax=b is consistent for some given b, and let p be a
solution. Then the solutions set of Ax=b is the set of all vectors of the form w = p + vh,
where vh is any solution of the... Warning: only applies to an equation Ax = b that has at
least one nonzero solution p. When Ax= b has no solution, the solution set is empty. -
Answer-homogeneous equation Ax = 0.
An indexed set of vectors {v1, ..., vp} in R^n is said to be _______ if the vector equation
x1v1 + x2v2 + ... + xpvp = 0 has only the trivial solution. - Answer-linearly independent
The set of vectors {v1, ..., vp} is said to be _______ if there exists weights c1, ..., cp, not
all zero, such that c1v1 + c2v2 + ... + cpvp = 0 - Answer-linearly dependent
The columns of a matrix A are linearly independent if and only if the equation Ax= 0 has
only the _____ - Answer-trivial solution
A set of two vectors {v1, v2} is ________ if at least one of the vectors is a multiple of the
other. - Answer-linearly dependent
A set of two vectors {v1, v2} is ________ if and only if neither of the vectors is a multiple
of the other. - Answer-linearly independent
Theorem: If a set contains more vectors than there are entries in each vector, then the
set is _______. That is, any set {v1, ..., vp} in R^n is _______ if p > n. - Answer-linearly
dependent
Theorem: If a set S = {v1, ..., vp} in R^n contains the zero vector, then the set is
_______. - Answer-linearly dependent
A _______ (or function or mapping) T from R^n to R^m is a rule that assigns to each
vector x in R^n a vector T(x) in R^m. The set R^n is called the ____ of T, and R^m is
called the ____ of T. - Answer-transformation, domain, codomain
The transformation T: R^2 to R^2 defined by T(x) = Ax is called a ______ transformation
- Answer-shear
