True or False

Question 1

A vector is any element of a vector space.

Answer 1

TRUE

Question 2

A vector space must contain at least two vectors.

Answer 2

FALSE

Question 3

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

Answer 3

FALSE

Question 4

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 4

FALSE

Question 5

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

Answer 5

TRUE

Question 6

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

Answer 6

FALSE

Question 7

Every subspace of a vector space is itself a vector space.

Answer 7

TRUE

Question 8

Every vector space is a subspace of itself.

Answer 8

TRUE

Question 9

Every subset of a vector space 𝑉 that contains the zero vector in 𝑉 is a subspace of 𝑉.

Answer 9

FALSE

Question 10

The kernel of a matrix transformation 𝑇𝐴 ∶ 𝑅n →𝑅m is a
subspace of 𝑅m.

Answer 10

FALSE

Question 11

The solution set of a consistent linear system 𝐴x = b of m equations in n unknowns is a subspace of 𝑅n.

Answer 11

FALSE

Question 12

The intersection of any two subspaces of a vector space 𝑉
is a subspace of 𝑉.

Answer 12

TRUE

Question 13

The union of any two subspaces of a vector space 𝑉 is a subspace of 𝑉.

Answer 13

FALSE

Question 14

The set of upper triangular n × n matrices is a subspace of
the vector space of all n × n matrices.

Answer 14

TRUE

Question 15

An expression of the form k1v1 + k2v2 + ⋅ ⋅ ⋅ krvr is called a linear combination.

Answer 15

TRUE

Question 16

The span of a single vector in 𝑅2 is a line.

Answer 16

FALSE

Question 17

The span of two vectors in 𝑅3 is a plane.

Answer 17

FALSE

Question 18

The span of a nonempty set 𝑆 of vectors in 𝑉 is the smallest subspace of 𝑉 that contains 𝑆.

Answer 18

TRUE

Question 19

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

Answer 19

TRUE

Question 20

Two subsets of a vector space 𝑉 that span the same subspace of 𝑉 must be equal.

Answer 20

FALSE

Question 21

The polynomials x − 1, (x − 1)2, and (x − 1)3 span 𝑃3.

Answer 21

FALSE

Question 22

A set containing a single vector is linearly independent.

Answer 22

FALSE

Question 23

No linearly independent set contains the zero vector.

Answer 23

TRUE

Question 24

Every linearly dependent set contains the zero vector.

Answer 24

FALSE

Question 25

If the set of vectors {v1, v2, v3} is linearly independent, then {kv1, kv2, kv3} is also linearly independent for every nonzero scalar k.

Answer 25

TRUE

Question 26

If v1, . . . , vn are linearly dependent nonzero vectors, then at least one vector vk is a unique linear combination of v1, . . . , vk−1.

Answer 26

TRUE

Question 27

The set of 2 × 2 matrices that contain exactly two 1’s and two 0’s is a linearly independent set in 𝑀22

Answer 27

FALSE

Question 28

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

Answer 28

TRUE

Question 29

The functions 𝑓1 and 𝑓2 are linearly dependent if there is a real number x such that k1𝑓1(x) + k2𝑓2(x) = 0 for some scalars k1 and k2.

Answer 29

FALSE

Question 30

If 𝑉 = span{v1, . . . , vn}, then {v1, . . . , vn} is a basis for 𝑉

Answer 30

FALSE

Question 31

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

Answer 31

FALSE

Question 32

If {v1, v2, . . . , vn} is a basis for a vector space 𝑉, then every vector in 𝑉 can be expressed as a linear combination of v1, v2, . . . , vn

Answer 32

TRUE

Question 33

The coordinate vector of a vector x in 𝑅n relative to the standard basis for 𝑅n is x

Answer 33

TRUE

Question 34

Every basis of 𝑃4 contains at least one polynomial of degree 3 or less

Answer 34

FALSE

Question 35

The zero vector space has dimension zero.

Answer 35

TRUE

Question 36

There is a set of 17 linearly independent vectors in 𝑅17

Answer 36

TRUE

Question 37

There is a set of 11 vectors that span 𝑅17

Answer 37

FALSE

Question 38

Every linearly independent set of five vectors in 𝑅5 is a basis for 𝑅5

Answer 38

TRUE

Question 39

Every set of five vectors that spans 𝑅5 is a basis for 𝑅5

Answer 39

TRUE

Question 40

Every set of vectors that spans 𝑅n contains a basis for 𝑅n

Answer 40

TRUE

Question 41

Every linearly independent set of vectors in 𝑅n is contained in some basis for 𝑅n

Answer 41

TRUE

Question 42

There is a basis for 𝑀22 consisting of invertible matrices

Answer 42

TRUE

Question 43

If 𝐴 has size n × n and 𝐼n, 𝐴, 𝐴2, . . . , 𝐴n2 are distinct matrices, then {𝐼n, 𝐴, 𝐴2, . . . , 𝐴n2 } is a linearly dependent set

Answer 43

TRUE

Question 44

There are at least two distinct three-dimensional subspaces of 𝑃2

Answer 44

FALSE

Question 45

There are only three distinct two-dimensional subspaces of 𝑃2.

Answer 45

FALSE

Question 46

If 𝐵1 and 𝐵2 are bases for a vector space 𝑉, then there exists a transition matrix from 𝐵1 to 𝐵2

Answer 46

TRUE

Question 47

Transition matrices are invertible

Answer 47

TRUE

Question 48

If 𝐵 is a basis for a vector space 𝑅n, then 𝑃𝐵→𝐵 is the identity matrix

Answer 48

TRUE

Question 49

If 𝑃𝐵1→𝐵2 is a diagonal matrix, then each vector in 𝐵2 is a scalar multiple of some vector in 𝐵1

Answer 49

TRUE

Question 50

If each vector in 𝐵2 is a scalar multiple of some vector in 𝐵1, then 𝑃𝐵1→𝐵2 is a diagonal matrix

Answer 50

FALSE

Question 51

If 𝐴 is a square matrix, then 𝐴 = 𝑃𝐵1→𝐵2 for some bases 𝐵1 and 𝐵2 for 𝑅n

Answer 51

FALSE

Question 52

The span of v1, . . . , vn is the column space of the matrix whose column vectors are v1, . . . , vn.

Answer 52

TRUE

Question 53

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

Answer 53

FALSE

Question 54

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 54

FALSE

Question 55

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

Answer 55

FALSE

Question 56

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

Answer 56

FALSE

Question 57

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 57

TRUE

Question 58

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 58

TRUE

Question 59

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 59

FALSE

Question 60

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

Answer 60

TRUE

Question 61

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

Answer 61

FALSE

Question 62

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

Answer 62

FALSE

Question 63

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

Answer 63

TRUE

Question 64

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

Answer 64

FALSE

Question 65

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

Answer 65

FALSE

Question 66

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

Answer 66

TRUE

Question 67

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

Answer 67

FALSE

Question 68

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 68

FALSE

Question 69

If rank(𝐴𝑇) = rank(𝐴), then 𝐴 is square

Answer 69

FALSE

Question 70

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

Answer 70

TRUE

Question 71

If 𝑉 is a subspace of 𝑅n and 𝑊 is a subspace of 𝑉, then 𝑊⟂ is a subspace of 𝑉⟂

Answer 71

FALSE

Question 72

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

Answer 72

FALSE

Question 73

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

Answer 73

FALSE

Question 74

If the characteristic polynomial of a matrix 𝐴 is p(𝜆) = 𝜆2 + 1 then 𝐴 is invertible

Answer 74

TRUE

Question 75

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 75

FALSE

Question 76

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

Answer 76

FALSE

Question 77

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

Answer 77

FALSE

Question 78

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

Answer 78

FALSE

Question 79

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

Answer 79

TRUE

Question 80

If 𝐴 and 𝐵 are similar n × n matrices, then there exists an invertible n × n matrix 𝑃 such that 𝑃𝐴 = 𝐵𝑃

Answer 80

TRUE

Question 81

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

Answer 81

FALSE

Question 82

If 𝐴 is diagonalizable and invertible, then 𝐴−1 is diagonalizable

Answer 82

TRUE

Question 83

If 𝐴 is diagonalizable, then 𝐴𝑇 is diagonalizable

Answer 83

TRUE

Question 84

If there is a basis for 𝑅n consisting of eigenvectors of an n × n matrix 𝐴, then 𝐴 is diagonalizable

Answer 84

TRUE

Question 85

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

Answer 85

TRUE

Question 86

If 0 is an eigenvalue of a matrix 𝐴, then 𝐴2 is singular

Answer 86

TRUE

Question 87

The matrix [1 0 0 1 0 0] is orthogonal

Answer 87

FALSE

Question 88

The matrix [1 −2 2 1] is orthogonal

Answer 88

FALSE

Question 89

An m × n matrix 𝐴 is orthogonal if 𝐴𝑇𝐴 = 𝐼

Answer 89

FALSE

Question 90

A square matrix whose columns form an orthogonal set is orthogonal

Answer 90

FALSE

Question 91

Every orthogonal matrix is invertible

Answer 91

TRUE

Question 92

If 𝐴 is an orthogonal matrix, then 𝐴2 is orthogonal and (det 𝐴)2 = 1

Answer 92

TRUE

Question 93

Every eigenvalue of an orthogonal matrix has absolute value 1

Answer 93

TRUE

Question 94

If 𝐴 is a square matrix and ‖𝐴u‖ = 1 for all unit vectors u, then 𝐴 is orthogonal

Answer 94

TRUE

Question 95

If 𝐴 is a square matrix, then 𝐴𝐴𝑇 and 𝐴𝑇𝐴 are orthogonally diagonalizable

Answer 95

TRUE

Question 96

If v1 and v2 are eigenvectors from distinct eigenspaces of a symmetric matrix with real entries, then ‖v1 + v2‖2 = ‖v1‖2 + ‖v2‖2

Answer 96

TRUE

Question 97

Every orthogonal matrix is orthogonally diagonalizable

Answer 97

FALSE

Question 98

If 𝐴 is both invertible and orthogonally diagonalizable, then 𝐴−1 is orthogonally diagonalizable

Answer 98

TRUE

Question 99

Every eigenvalue of an orthogonal matrix has absolute value 1.

Answer 99

TRUE

Question 100

If 𝐴 is an n × n orthogonally diagonalizable matrix, then there exists an orthonormal basis for 𝑅n consisting of eigenvectors of 𝐴.

Answer 100

TRUE

Question 101

If 𝐴 is orthogonally diagonalizable, then 𝐴 has real eigenvalues.

Answer 101

TRUE

Question 102

If all eigenvalues of a symmetric matrix 𝐴 are positive, then 𝐴 is positive definite

Answer 102

TRUE

Question 103

x2^1− x2^2 + x3^2 + 4x1x2x3 is a quadratic form

Answer 103

FALSE

Question 104

(x1 − 3x2)^2 is a quadratic form

Answer 104

TRUE

Question 105

A positive definite matrix is invertible

Answer 105

TRUE

Question 106

A symmetric matrix is either positive definite, negative definite, or indefinite

Answer 106

FALSE

Question 107

If 𝐴 is positive definite, then −𝐴 is negative definite

Answer 107

TRUE

Question 108

x · x is a quadratic form for all x in 𝑅n

Answer 108

TRUE

Question 109

If 𝐴 is symmetric and invertible, and if x𝑇𝐴x is a positive definite quadratic form, then x𝑇𝐴−1x is also a positive definite quadratic form

Answer 109

TRUE

Question 110

If 𝐴 is symmetric and has only positive eigenvalues, then x𝑇𝐴x is a positive definite quadratic form

Answer 110

TRUE

Question 111

If 𝐴 is a 2 × 2 symmetric matrix with positive entries and det(𝐴) > 0, then 𝐴 is positive definite

Answer 111

TRUE

Question 112

If 𝐴 is symmetric, and if the quadratic form x𝑇𝐴x has no cross product terms, then 𝐴 must be a diagonal matrix

Answer 112

TRUE

Question 113

If x𝑇𝐴x is a positive definite quadratic form in two variables and c ≠ 0, then the graph of the equation x𝑇𝐴x = c is an ellipse

Answer 113

FALSE