Homework True or False

Question 1

A linear system whose equations are all homogeneous must be consistent.

Answer 1

True

Question 2

Multiplying a row of an augmented matrix through by zero is an acceptable elementary
row operation

Answer 2

False

Question 3

The linear system
𝑥 − 𝑦 = 3
2𝑥 − 2𝑦 = 𝑘
cannot have a unique solution, regardless of the value of 𝑘.

Answer 3

True

Question 4

A single linear equation with two or more unknowns must have infinitely many solutions

Answer 4

True

Question 5

If the number of equations in a linear system exceeds the number of unknowns, then the
system must be inconsistent.

Answer 5

False

Question 6

If each equation in a consistent linear system is multiplied through by a constant 𝑐, then
all solutions to the new system can be obtained by multiplying solutions from the original
system by 𝑐.

Answer 6

False

Question 7

Elementary row operations permit one row of an augmented matrix to be subtracted from
another.

Answer 7

True

Question 8

The linear system with corresponding augmented matrix
[
2 −1 4
0 0 −1
]
is consistent.

Answer 8

False

Question 9

If a matrix is in reduced row echelon form, then it is also in row echelon form.

Answer 9

True

Question 10

If an elementary row operation is applied to a matrix that is in row echelon form, the
resulting matrix will still be in row echelon form.

Answer 10

False

Question 11

Every matrix has a unique row echelon form.

Answer 11

False

Question 12

A homogeneous linear system in n unknowns whose corresponding augmented matrix
has a reduced row echelon form with 𝑟 leading 1’s has 𝑛 − 𝑟 free variables.

Answer 12

True

Question 13

All leading 1’s in a matrix in row echelon form must occur in different columns.

Answer 13

True

Question 14

If every column of a matrix in row echelon form has a leading 1, then all entries that are
not leading 1’s are zero.

Answer 14

False

Question 15

If a homogeneous linear system of 𝑛 equations in 𝑛 unknowns has a corresponding
augmented matrix with a reduced row echelon form containing 𝑛 leading 1’s, then the
linear system has only the trivial solution.

Answer 15

True

Question 16

If the reduced row echelon form of the augmented matrix for a linear system has a row of
zeros, then the system must have infinitely many solutions.

Answer 16

False

Question 17

If a linear system has more unknowns than equations, then it must have infinitely many
solutions.

Answer 17

False

Question 18

The matrix [
1 2 3
4 5 6
] has no main diagonal.

Answer 18

True

Question 19

For every matrix 𝐴, it is true that (𝐴^𝑇)^𝑇 = 𝐴.

Answer 19

True

Question 20

For every square matrix 𝐴, it is true that 𝑡𝑟(𝐴^𝑇) = 𝑡𝑟(𝐴).

Answer 20

True

Question 21

If 𝐴 is an 𝑛 × 𝑛 matrix and 𝑐 is a scalar, then 𝑡𝑟(𝑐𝐴) = 𝑐 𝑡𝑟(𝐴).

Answer 21

True

Question 22

If 𝐴, 𝐵, and 𝐶 are matrices of the same size such that 𝐴 − 𝐶 = 𝐵 − 𝐶, then 𝐴 = 𝐵.

Answer 22

True

Question 23

An 𝑚 × 𝑛 matrix has 𝑚 column vectors and 𝑛 row vectors.

Answer 23

False

Question 24

If 𝐴 and 𝐵 are 2 × 2 matrices, then 𝐴𝐵 = 𝐵𝐴.

Answer 24

False

Question 25

The ith row vector of a matrix product 𝐴𝐵 can be computed by multiplying 𝐴 by the
ith row vector of 𝐵.

Answer 25

False

Question 26

If 𝐴 and 𝐵 are square matrices of the same order, then
𝑡𝑟(𝐴𝐵) = 𝑡𝑟(𝐴)𝑡𝑟(𝐵)

Answer 26

False

Question 27

If 𝐴 and 𝐵 are square matrices of the same order, then
(𝐴𝐵)^𝑇 = 𝐴^𝑇 𝐵

Answer 27

False

Question 28

If 𝐴 is a 6 × 4 matrix and 𝐵 is an 𝑚 × 𝑛 matrix such that 𝐵^𝑇 𝐴^𝑇
is a 2 × 6 matrix,
then 𝑚 = 4 and 𝑛 = 2.

Answer 28

True

Question 29

If 𝐴, 𝐵, and 𝐶 are square matrices of the same order such that 𝐴𝐶 = 𝐵𝐶, then 𝐴 = 𝐶

Answer 29

False

Question 30

If 𝐴𝐵 + 𝐵𝐴 is defined, then 𝐴 and 𝐵 are square matrices of the same size.

Answer 30

True

Question 31

If 𝐵 has a column of zeros, then so does 𝐴𝐵 if this product is defined.

Answer 31

True

Question 32

If 𝐵 has a column of zeros, then so does 𝐵𝐴 if this product is defined.

Answer 32

False

Question 33

Two 𝑛 × 𝑛 matrices, 𝐴 and 𝐵, are inverses of one another if and only if
𝐴𝐵 = 𝐵𝐴 = 0.

Answer 33

False

Question 34

For all square matrices 𝐴 and 𝐵 of the same size, it is true that (𝐴 + 𝐵)^2 = 𝐴^2 +2𝐴𝐵 + 𝐵^2

Answer 34

False

Question 35

For all square matrices 𝐴 and 𝐵 of the same size, it is true that 𝐴
2 − 𝐵2 =
(𝐴 − 𝐵)(𝐴 + 𝐵)

Answer 35

False

Question 36

If 𝐴 and 𝐵 are invertible matrices of the same size, then 𝐴𝐵 is invertible and (𝐴𝐵)^−1 = 𝐴^−1 𝐵^−1

Answer 36

False

Question 37

If 𝐴 and 𝐵 are matrices such that 𝐴𝐵 is defined, then it is true that
(𝐴𝐵)^𝑇 = 𝐴^𝑇 𝐵^𝑇

Answer 37

False

Question 38

The matrix [
𝑎 𝑏
𝑐 𝑑
] is invertible if and only if
𝑎𝑑 − 𝑏𝑐 ≠ 0.

Answer 38

True

Question 39

If 𝐴 and 𝐵 are matrices of the same size and 𝑘 is a constant, then
(𝑘𝐴 + 𝐵)^𝑇 = 𝑘𝐴^𝑇 + 𝐵^𝑇

Answer 39

True

Question 40

If 𝐴 is an invertible matrix, then so is 𝐴^𝑇

Answer 40

True

Question 41

If
𝑝(𝑥) = 𝑎0 + 𝑎1𝑥 + 𝑎2𝑥^2 + ⋯ + 𝑎_𝑚 𝑥^𝑚
and 𝐼 is an identity matrix, then
𝑝(𝐼) = 𝑎0 + 𝑎1 + 𝑎2 + ⋯ + 𝑎𝑚.

Answer 41

False

Question 42

A square matrix containing a row or column of zeros cannot be invertible

Answer 42

True

Question 43

The sum of two invertible matrices of the same size must be invertible.

Answer 43

False

Question 44

The product of two elementary matrices of the same size must be an elementary matrix.

Answer 44

False

Question 45

Every elementary matrix is invertible.

Answer 45

True

Question 46

f 𝐴 and 𝐵 are row equivalent, and if 𝐵 and 𝐶 are row equivalent, then 𝐴 and 𝐶 are row
equivalent.

Answer 46

True

Question 47

f 𝐴 is an 𝑛 × 𝑛 matrix that is not invertible, then the linear system 𝐴𝒙 = 0 has infinitely
many solutions.

Answer 47

True

Question 48

If 𝐴 is an 𝑛 × 𝑛 matrix that is not invertible, then the matrix obtained by interchanging
two rows of 𝐴 cannot be invertible.

Answer 48

True

Question 49

If 𝐴 is invertible and a multiple of the first row of 𝐴 is added to the second row, then the
resulting matrix is invertible.

Answer 49

True

Question 50

An expression of an invertible matrix 𝐴 as a product of elementary matrices is unique

Answer 50

False

Question 51

It is impossible for a system of linear equations to have exactly two solutions.

Answer 51

True

Question 52

If 𝐴 is a square matrix, and if the linear system 𝐴𝒙 = 𝒃 has a unique solution, then the
linear system 𝐴𝒙 = 𝒄 also must have a unique solution.

Answer 52

True

Question 53

If 𝐴 and 𝐵 are 𝑛 × 𝑛 matrices such that 𝐴𝐵 = 𝐼_𝑛, then 𝐵𝐴 = 𝐼_𝑛.

Answer 53

True

Question 54

If 𝐴 and 𝐵 are row equivalent matrices, then the linear systems 𝐴𝒙 = 𝟎 and 𝐵𝒙 = 𝟎
have the same solution set

Answer 54

True

Question 55

Let 𝐴 be an 𝑛 × 𝑛 matrix and 𝑆 is an 𝑛 × 𝑛 invertible matrix. If 𝒙 is a solution to
system (𝑆^−1 𝐴 𝑆)𝒙 = 𝒃, then 𝑆𝒙 is a solution system 𝐴𝒚 = 𝑆𝒃.

Answer 55

True

Question 56

Let 𝐴 be an 𝑛 × 𝑛 matrix. The linear system 𝐴𝒙 = 4𝒙 has a unique solution if and only
if 𝐴 − 4𝐼 is an invertible matrix.

Answer 56

True

Question 57

Let 𝐴 and 𝐵 be 𝑛 × 𝑛 matrices. If 𝐴 or 𝐵 (or both) are not invertible, then neither is 𝐴𝐵.

Answer 57

True

Question 58

Let 𝐴 and 𝐵 be 𝑛 × 𝑛 matrices. If 𝐴 or 𝐵 (or both) are not invertible, then neither is 𝐴𝐵.

Answer 58

True

Question 59

The transpose of an upper triangular matrix is an upper triangular matrix.

Answer 59

False

Question 60

The sum of an upper triangular matrix and a lower triangular matrix is a diagonal matrix.

Answer 60

False

Question 61

All entries of a symmetric matrix are determined by the entries occurring on and above
the main diagonal.

Answer 61

True

Question 62

All entries of an upper triangular matrix are determined by the entries occurring on and
above the main diagonal.

Answer 62

True

Question 63

The inverse of an invertible lower triangular matrix is an upper triangular matrix

Answer 63

False

Question 64

A diagonal matrix is invertible if and only if all of its diagonal entries are positive.

Answer 64

False

Question 65

The sum of a diagonal matrix and a lower triangular matrix is a lower triangular matrix.

Answer 65

True

Question 66

𝐴 matrix that is both symmetric and upper triangular must be a diagonal matrix

Answer 66

True

Question 67

If 𝐴 and 𝐵 are 𝑛 × 𝑛 matrices such that 𝐴 + 𝐵 is symmetric, then 𝐴 and 𝐵 are
symmetric

Answer 67

False

Question 68

If 𝐴 and 𝐵 are 𝑛 × 𝑛 matrices such that 𝐴 + 𝐵 is upper triangular, then 𝐴 and 𝐵 are
upper triangular.

Answer 68

False

Question 69

f 𝐴^2 is a symmetric matrix, then 𝐴 is a symmetric matrix

Answer 69

False

Question 70

If 𝑘𝐴 is a symmetric matrix for some 𝑘 ≠ 0, then 𝐴 is a symmetric matrix.

Answer 70

True

Question 71

The determinant of the 2 × 2 matrix [
𝑎 𝑏
𝑐 𝑑 ] is 𝑎𝑑 + 𝑏𝑐

Answer 71

False

Question 72

Two square matrices that have the same determinant must have the same size.

Answer 72

False

Question 73

The minor 𝑀_𝑖𝑗 is the same as the cofactor 𝐶_𝑖𝑗 if 𝑖 + 𝑗 is even.

Answer 73

True

Question 74

If 𝐴 is a 3 × 3 symmetric matrix, then 𝐶𝑖𝑗 = 𝐶𝑗𝑖 for all 𝑖 and 𝑗.

Answer 74

True

Question 75

The number obtained by a cofactor expansion of a matrix 𝐴 is independent of the row or
column chosen for the expansion

Answer 75

True

Question 76

If 𝐴 is a square matrix whose minors are all zero, then 𝑑𝑒𝑡(𝐴) = 0.

Answer 76

True

Question 77

The determinant of a lower triangular matrix is the sum of the entries along the main
diagonal.

Answer 77

False

Question 78

For every square matrix 𝐴 and every scalar 𝑐, it is true that 𝑑𝑒𝑡 (𝑐𝐴) = 𝑐 𝑑𝑒𝑡(𝐴)

Answer 78

False

Question 79

For all square matrices 𝐴 and 𝐵, it is true that
det(𝐴 + 𝐵) = det(𝐴) + det (𝐵)

Answer 79

False

Question 80

For every 2 × 2 matrix 𝐴 it is true that
det(𝐴^2) = (det(𝐴))^2

Answer 80

True

Question 81

f 𝐴 is a 4 × 4 matrix and 𝐵 is obtained from 𝐴 by interchanging the first two rows and
then interchanging the last two rows, then 𝑑𝑒𝑡(𝐵) = 𝑑𝑒𝑡(𝐴).

Answer 81

True

Question 82

If 𝐴 is a 3 × 3 matrix and 𝐵 is obtained from 𝐴 by multiplying the first column by 4 and
multiplying the third column by 3/4
, then det(𝐵) = 3 det(𝐴).

Answer 82

True

Question 83

If 𝐴 is a 3 × 3 matrix and 𝐵 is obtained from 𝐴 by adding 5 times the first row to each
of the second and third rows, then det(𝐵) = 25 det(𝐴).

Answer 83

False

Question 84

If 𝐴 is an 𝑛 × 𝑛 matrix and 𝐵 is obtained from 𝐴 by multiplying each row of 𝐴 by its
row number, then
det(𝐵) = 𝑛(𝑛 + 1) / 2 * det (𝐴)

Answer 84

False

Question 85

If 𝐴 is a square matrix with two identical columns, then det(𝐴) = 0

Answer 85

True

Question 86

If the sum of the second and fourth row vectors of a 6 × 6 matrix 𝐴 is equal to the last
row vector, then 𝑑𝑒𝑡(𝐴) = 0.

Answer 86

True

Question 87

If 𝐴 is a 3 × 3 matrix, then det(2𝐴) = 2 det(𝐴)

Answer 87

False

Question 88

If 𝐴 and 𝐵 are square matrices of the same size such that det(𝐴) = det(𝐵),
then det(𝐴 + 𝐵) = 2 det(𝐴)

Answer 88

False

Question 89

If 𝐴 and 𝐵 are square matrices of the same size and 𝐴 is invertible, then
det(𝐴^−1𝐵𝐴) = det (𝐵)

Answer 89

True

Question 90

A square matrix 𝐴 is invertible if and only if det(𝐴) = 0.

Answer 90

False

Question 91

If 𝐴 is a square matrix and the linear system 𝐴𝒙 = 𝟎 has multiple solutions for 𝒙, then
𝑑𝑒𝑡(𝐴) = 0.

Answer 91

True

Question 92

If 𝐴 is an 𝑛 × 𝑛 matrix and there exists an 𝑛 × 1 matrix 𝒃 such that the linear system
𝐴𝒙 = 𝒃 has no solutions, then the reduced row echelon form of 𝐴 cannot be 𝐼_𝑛.

Answer 92

True

Question 93

If 𝐸 is an elementary matrix, then 𝐸𝒙 = 𝟎 has only the trivial solution

Answer 93

True

Question 94

If 𝐴 is an invertible matrix, then the linear system 𝐴𝒙 = 𝟎 has only the trivial solution if and
only if the linear system 𝐴^−1 𝒙 = 𝟎 has only the trivial solution.

Answer 94

True

Question 95

Two equivalent vectors must have the same initial point.

Answer 95

False

Question 96

The vectors (𝑎, 𝑏) and (𝑎, 𝑏, 0) are equivalent.

Answer 96

False

Question 97

If 𝑘 is a scalar and 𝒗 is a vector, then 𝒗 and 𝑘𝒗 are parallel if and only if 𝑘 ≥ 0.

Answer 97

False

Question 98

The vectors 𝒗 + (𝒖 + 𝒘) and (𝒘 + 𝒗) + 𝒖 are the same.

Answer 98

True

Question 99

If 𝒖 + 𝒗 = 𝒖 + 𝒘, then 𝒗 = 𝒘.

Answer 99

True

Question 100

If 𝑎 and 𝑏 are scalars such that 𝑎𝒖 + 𝑏𝒗 = 0, then 𝒖 and 𝒗 are parallel vectors.

Answer 100

False

Question 101

Collinear vectors with the same length are equal.

Answer 101

False

Question 102

If (𝑎, 𝑏, 𝑐) + (𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 𝑧), then (𝑎, 𝑏, 𝑐) must be the zero vector.

Answer 102

True

Question 103

If 𝑘 and 𝑚 are scalars and 𝒖 and 𝒗 are vectors, then
(𝑘 + 𝑚)(𝒖 + 𝒗) = 𝑘𝒖 + 𝑚𝒗

Answer 103

False

Question 104

If the vectors 𝒗 and 𝒘 are given, then the vector equation
3(2𝒗 − 𝒙) = 5𝒙 − 4𝒘 + 𝒗
can be solved for 𝒙.

Answer 104

True

Question 105

The linear combinations 𝑎1𝒗𝟏 + 𝑎2𝒗𝟐 and 𝑏1𝒗𝟏 + 𝑏2𝒗𝟐 can only be equal if 𝑎1 = 𝑏1
and 𝑎2 = 𝑏2.

Answer 105

False

Question 106

If each component of a vector in 𝑅^3 is doubled, the norm of that vector is doubled.

Answer 106

True

Question 107

In 𝑅^2, the vectors of norm 5 whose initial points are at the origin have terminal points
lying on a circle of radius 5 centred at the origin.

Answer 107

True

Question 108

Every vector in 𝑅^𝑛 has a positive norm

Answer 108

False

Question 109

f 𝒗 is a nonzero vector in 𝑅^𝑛, there are exactly two unit vectors that are parallel to 𝒗.

Answer 109

True

Question 110

f ‖𝒖‖ = 2, ‖𝒗‖ = 1, and 𝒖 ⋅ 𝒗 = 1, then the angle between 𝒖 and 𝒗 is 𝜋/3 radians.

Answer 110

True

Question 111

The expressions (𝒖 ⋅ 𝒗) + 𝒘 and 𝒖 ⋅ (𝒗 + 𝒘) are both meaningful and equal to each
other.

Answer 111

False

Question 112

f 𝒖 ⋅ 𝒗 = 𝒖 ⋅ 𝒘, then 𝒗 = 𝒘.

Answer 112

False

Question 113

If 𝒖 ⋅ 𝒗 = 0, then either 𝒖 = 0 𝑜𝑟 𝒗 = 0.

Answer 113

False

Question 114

n 𝑅^2, if 𝒖 lies in the first quadrant and 𝒗 lies in the third quadrant, then 𝒖 ⋅ 𝒗 cannot be
positive.

Answer 114

True

Question 115

For all vectors 𝒖, 𝒗, 𝑎𝑛𝑑 𝒘 in 𝑅
𝑛, we have
‖𝒖 + 𝒗 + 𝒘‖ ≤ ‖𝒖‖ + ‖𝒗‖ + ‖𝒘‖

Answer 115

True

Question 116

The vectors (3, −1, 2) and (0, 0, 0) are orthogonal.

Answer 116

True

Question 117

If 𝒖 and 𝒗 are orthogonal vectors, then for all nonzero scalars 𝑘 and 𝑚, 𝑘𝒖 and 𝑚𝒗 are
orthogonal vectors.

Answer 117

True

Question 118

The orthogonal projection of 𝒖 on 𝒂 is perpendicular to the vector component of 𝒖
orthogonal to 𝒂.

Answer 118

True

Question 119

If 𝒂 and 𝒃 are orthogonal vectors, then for every nonzero vector 𝒖, we have
𝑝𝑟𝑜𝑗_𝑎(𝑝𝑟𝑜𝑗_𝑏(𝒖)) = 𝟎

Answer 119

True

Question 120

If 𝒂 and 𝒖 are nonzero vectors, then
𝑝𝑟𝑜𝑗_𝑎(𝑝𝑟𝑜𝑗_𝑎(𝒖)) = 𝑝𝑟𝑜𝑗_𝑎(𝒖)

Answer 120

True

Question 121

If the relationship
𝑝𝑟𝑜𝑗𝑎 𝒖 = 𝑝𝑟𝑜𝑗_𝑎 𝒗
holds for some nonzero vector 𝒂, then 𝒖 = 𝒗.

Answer 121

False

Question 122

For all vectors 𝒖 and 𝒗, it is true that
‖𝒖 + 𝒗‖ = ‖𝒖‖ + ‖𝒗‖

Answer 122

False

Question 123

The cross product of two nonzero vectors 𝒖 and 𝒗 is a nonzero vector if and only if 𝒖 and
𝒗 are not parallel.

Answer 123

True

Question 124

A normal vector to a plane can be obtained by taking the cross product of two nonzero
and noncollinear vectors lying in the plane.

Answer 124

True

Question 125

The scalar triple product of 𝒖, 𝒗 and 𝒘 determines a vector whose length is equal to the
volume of the parallelepiped determined by 𝒖, 𝒗 and 𝒘.

Answer 125

False

Question 126

If 𝒖 and 𝒗 are vectors in 3‐space, then ‖𝒗 × 𝒖‖ is equal to the area of the parallelogram
determined by 𝒖 and 𝒗.

Answer 126

True

Question 127

For all vectors 𝒖, 𝒗 and 𝒘 in 3-space, the vectors (𝒖 × 𝒗) × 𝒘 and 𝒖 × (𝒗 × 𝒘) are the
same.

Answer 127

False

Question 128

If 𝒖, 𝒗 and 𝒘 are vectors in 𝑅
3, where 𝒖 is nonzero and 𝒖 × 𝒗 = 𝒖 × 𝒘, then 𝒗 = 𝒘

Answer 128

False

Question 129

A vector is any element of a vector space.

Answer 129

True

Question 130

A vector space must contain at least two vectors.

Answer 130

False

Question 131

The set of positive real numbers is a vector space if vector addition and scalar
multiplication are the usual operations of addition and multiplication of real numbers.

Answer 131

False

Question 132

If 𝒖 is a vector and 𝑘 is a scalar such that 𝑘𝒖 = 𝟎, then it must be true that 𝑘 = 0.

Answer 132

False

Question 133

In every vector space the vectors (−1)𝒖 𝑎𝑛𝑑 − 𝒖 are the same.

Answer 133

True

Question 134

n the vector space 𝐹(−∞, ∞) any function whose graph passes through the origin is a
zero vector

Answer 134

False

Question 135

An expression of the form 𝑘_1 𝒗_𝟏 + 𝑘_2 𝒗_𝟐 + ⋯ + 𝑘_𝑟 𝒗_𝒓
is called a linear combination.

Answer 135

True

Question 136

The span of a single vector in 𝑅^2
is a line.

Answer 136

True

Question 137

The span of two vectors in 𝑅^3
is a plane.

Answer 137

False

Question 138

The span of any finite set of vectors in a vector space is closed under addition and scalar
multiplication.

Answer 138

False

Question 139

A set containing a single vector is linearly independent.

Answer 139

False

Question 140

No linearly independent set contains the zero vector.

Answer 140

True

Question 141

Every linearly dependent set contains the zero vector.

Answer 141

False

Question 142

If the set of vectors {𝒗_𝟏, 𝒗_𝟐, 𝒗_𝟑} is linearly independent, then {𝑘𝒗𝟏, 𝑘 𝒗_𝟐, 𝑘 𝒗_𝟑} is also
linearly independent for every nonzero scalar 𝑘.

Answer 142

True

Question 143

If 𝒗_𝟏, …, 𝒗_𝒏 are linearly dependent nonzero vectors, then at least one vector 𝒗_𝒌 is a
unique linear combination of 𝒗_𝟏, … , 𝒗_𝒌−𝟏.

Answer 143

True

Question 144

The set of 2 × 2 matrices that contain exactly two 1’s and two 0’s is a linearly
independent set in 𝑀_22.

Answer 144

False

Question 145

The three polynomials (𝑥 − 1)(𝑥 + 2), 𝑥(𝑥 + 2), 𝑎𝑛𝑑 𝑥(𝑥 − 1) are linearly
independent.

Answer 145

True

Question 146

The functions 𝑓_1 and 𝑓_2 are linearly dependent if there is a real number 𝑥 such that
𝑘_1 𝑓_1(𝑥) + 𝑘_2 𝑓_2(𝑥) = 0 for some scalars 𝑘_1 and 𝑘_2.

Answer 146

False

Question 147

f 𝑉 = 𝑠𝑝𝑎𝑛{𝒗_𝟏, … , 𝒗_𝒏}, then {𝒗_𝟏, … , 𝒗_𝒏} is a basis for 𝑉.

Answer 147

False

Question 148

Every linearly independent subset of a vector space 𝑉 is a basis for 𝑉.

Answer 148

False

Question 149

f {𝒗_𝟏, …, 𝒗_𝒏} is a basis for a vector space 𝑉, then every vector in 𝑉 can be expressed as
a linear combination of 𝒗_𝟏, … , 𝒗_𝒏.

Answer 149

True

Question 150

The coordinate vector of a vector 𝒙 in 𝑅^𝑛 relative to the standard basis for 𝑅^𝑛 is 𝒙.

Answer 150

True

Question 151

Every basis of 𝑃_4 contains at least one polynomial of degree 3 or less.

Answer 151

False

Question 152

The zero vector space has dimension zero

Answer 152

True

Question 153

There is a set of 17 linearly independent vectors in 𝑅^17.

Answer 153

True

Question 154

There is a set of 11 vectors that span 𝑅^17.

Answer 154

False

Question 155

Every linearly independent set of five vectors in 𝑅^5 is a basis for 𝑅^5

Answer 155

True

Question 156

Every set of five vectors that spans 𝑅^5 is a basis for 𝑅^5

Answer 156

True

Question 157

Every set of vectors that spans 𝑅^𝑛 contains a basis for 𝑅^𝑛

Answer 157

True

Question 158

Every linearly independent set of vectors in 𝑅^𝑛 is contained in some basis for 𝑅^𝑛.

Answer 158

True

Question 159

If 𝐴 has size 𝑛 × 𝑛 and
𝐼_𝑛, 𝐴, 𝐴^2, … , 𝐴^𝑛^2
are distinct matrices, then
{𝐼_𝑛, 𝐴, 𝐴^2, … , 𝐴^𝑛^2} is
a linearly dependent set.

Answer 159

True

Question 160

The span of v_1, …, v_n is the column space of the matrix whose column vectors
are v_1, …, v_n.

Answer 160

True

Question 161

The column space of a matrix A is the set of solutions of Ax = b

Answer 161

Fasle

Question 162

If R is the reduced row echelon form of A, then those column vectors of R that contain
the leading 1’s form a basis for the column space of A.

Answer 162

False

Question 163

The set of nonzero row vectors of a matrix A is a basis for the row space of A

Answer 163

False

Question 164

If A and B are n × n matrices that have the same row space, then A and B have the same
column space

Answer 164

False

Question 165

If E is an m × m elementary matrix and A is an m × n matrix, then the null space of EA is
the same as the null space of A.

Answer 165

True

Question 166

If E is an m × m elementary matrix and A is an m × n matrix, then the row space of EA is
the same as the row space of A.

Answer 166

True

Question 167

If E is an m × m elementary matrix and A is an m × n matrix, then the column space
of EA is the same as the column space of A.

Answer 167

False

Question 168

The system Ax = b is inconsistent if and only if b is not in the column space of A.

Answer 168

True

Question 169

There is an invertible matrix A and a singular matrix B such that the row spaces
of A and B are the same.

Answer 169

False

Question 170

Either the row vectors or the column vectors of a square matrix are linearly
independent

Answer 170

False

Question 171

A matrix with linearly independent row vectors and linearly independent column vectors
is square.

Answer 171

True

Question 172

The nullity of a nonzero m × n matrix is at most m

Answer 172

False

Question 173

Adding one additional column to a matrix increases its rank by one

Answer 173

False

Question 174

The nullity of a square matrix with linearly dependent rows is at least one

Answer 174

True

Question 175

If A is square and Ax = b is inconsistent for some vector b, then the nullity of A is zero.

Answer 175

False

Question 176

If a matrix A has more rows than columns, then the dimension of the row space is
greater than the dimension of the column space.

Answer 176

False

Question 177

If rank (A^T) = rank(A), then A is square.

Answer 177

False

Question 178

There is no 3 × 3 matrix whose row space and null space are both lines in 3-space.

Answer 178

True

Question 179

f 𝐴 is an 𝑚 × 𝑛 matrix, then the codomain of the transformation 𝑇_𝐴 is 𝑅^𝑛

Answer 179

False

Question 180

If 𝐴 is a 2 × 3 matrix, then the domain of the transformation 𝑇_𝐴 is 𝑅_2.

Answer 180

False

Question 181

If 𝐴 is a square matrix and 𝐴𝒙 = 𝜆𝒙 for some nonzero scalar 𝜆, then 𝒙 is an eigenvector
of 𝐴.

Answer 181

False

Question 182

If 𝜆 is an eigenvalue of a matrix 𝐴, then the linear system (𝜆𝐼 − 𝐴)𝒙 = 𝟎 has only the
trivial solution.

Answer 182

False

Question 183

If the characteristic polynomial of a matrix 𝐴 is
𝑝(𝜆) = 𝜆^2 + 1
then 𝐴 is invertible.

Answer 183

True

Question 184

If 𝜆 is an eigenvalue of a matrix 𝐴, then the eigenspace of 𝐴 corresponding to 𝜆 is the set
of eigenvectors of 𝐴 corresponding to 𝜆.

Answer 184

False

Question 185

The eigenvalues of a matrix 𝐴 are the same as the eigenvalues of the reduced row echelon
form of 𝐴.

Answer 185

False

Question 186

If 0 is an eigenvalue of a matrix 𝐴, then the set of columns of 𝐴 is linearly independent.

Answer 186

False

Question 187

An 𝑛 × 𝑛 matrix with fewer than 𝑛 distinct eigenvalues is not diagonalizable.

Answer 187

False

Question 188

An 𝑛 × 𝑛 matrix with fewer than 𝑛 linearly independent eigenvectors is not
diagonalizable.

Answer 188

True

Question 189

If 𝐴 is diagonalizable, then there is a unique matrix 𝑃 such that 𝑃
−1 𝐴𝑃 is diagonal.

Answer 189

False

Question 190

If every eigenvalue of a matrix 𝐴 has algebraic multiplicity 1, then 𝐴 is diagonalizable.

Answer 190

True

Question 191

The dot product on 𝑅^2
is an example of a weighted inner product.

Answer 191

True

Question 192

The inner product of two vectors cannot be a negative real number.

Answer 192

False

Question 193

〈𝒖, 𝒗 + 𝒘〉 = 〈𝒗, 𝒖〉 + 〈𝒘, 𝒖〉.

Answer 193

True

Question 194

〈𝑘𝒖, 𝑘𝒗〉 = 𝑘2〈𝒖, 𝒗〉.

Answer 194

True

Question 195

If 〈𝒖, 𝒗〉 = 0, then 𝒖 = 𝟎 or 𝒗 = 𝟎.

Answer 195

False

Question 196

f ‖𝒗‖^2 = 𝟎, then 𝒗 = 0.

Answer 196

True

Question 197

If 𝐴 is an 𝑛 × 𝑛 matrix, then 〈𝒖, 𝒗〉 = 𝐴𝒖 ⋅ 𝐴𝒗 defines an inner product on 𝑅^𝑛.

Answer 197

False

Question 198

If 𝒖 is orthogonal to every vector of a subspace 𝑊, then 𝒖 = 𝟎.

Answer 198

False

Question 199

f 𝒖 is a vector in both 𝑊 and 𝑊⊥, then 𝒖 = 𝟎.

Answer 199

True

Question 200

If 𝒖 and 𝒗 are vectors in 𝑊⊥, then 𝒖 + 𝒗 is in 𝑊⊥.

Answer 200

True

Question 201

If 𝒖 is a vector in 𝑊⊥ and 𝑘 is a real number, then 𝑘𝒖 is in 𝑊⊥.

Answer 201

True

Question 202

If 𝒖 and 𝒗 are orthogonal, then |〈𝒖, 𝒗〉| = ‖𝒖‖‖𝒗‖.

Answer 202

False

Question 203

If 𝒖 and 𝒗 are orthogonal, then ‖𝒖 + 𝒗‖ = ‖𝒖‖ + ‖𝒗‖.

Answer 203

False

Question 204

Every linearly independent set of vectors in an inner product space is orthogonal.

Answer 204

False

Question 205

Every orthogonal set of vectors in an inner product space is linearly independent.

Answer 205

False

Question 206

Every nontrivial subspace of 𝑅^3
has an orthonormal basis with respect to the Euclidean
inner product.

Answer 206

True

Question 207

Every nonzero finite‐dimensional inner product space has an orthonormal basis.

Answer 207

True

Question 208

𝑝𝑟𝑜𝑗_𝑊𝒙 is orthogonal to every vector of 𝑊.

Answer 208

False