True/false Matrices Midterm #2

Front 1

In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.

Back 1

true

Front 2

If A and B are n x n and invertible, then A^-1B^-1 is the inverse of AB.

Back 2

false, B^-1A^-1 is the inverse

Front 3

If A = [ a b
c d ] and ab - cd does not equal 0, then A is invertible.

Back 3

false, A is invertible if ad-cb does not equal 0.

Front 4

If A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in R^n.

Back 4

true

Front 5

Each elementary matrix is invertible.

Back 5

true

Front 6

A product of invertible n x n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.

Back 6

false, in reverse order

Front 7

If A is invertible, then the inverse of A^-1 is A itself.

Back 7

true

Front 8

If A = [ a b
c d ] and ad = bc, then A is not invertible.

Back 8

true

Front 9

If A can be row reduced to the identity matrix, then A must be invertible.

Back 9

true

Front 10

If A is invertible, then elementary row operations that reduce A to the identity In(eye-subn) also reduce A^-1 to In(eye-subn).

Back 10

false, it reduces In(eye subn) to A^-1

Front 11

If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.

Back 11

true

Front 12

If the columns of A span Rn, then the columns are linearly independent.

Back 12

true

Front 13

If A is an n x n matrix, then the equation Ax = b has at least one solution for each b in Rn.

Back 13

false, this is only true for invertible matrices.

Front 14

If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions.

Back 14

true

Front 15

If A transpose is not invertible, then A is not invertible.

Back 15

true

Front 16

If there is an n x n matrix D such that AD = I, then there is also an n x n matrix C such that CA = I.

Back 16

true

Front 17

If the columns of A are linearly independent, then the columns of A span Rn.

Back 17

true

Front 18

If the equation Ax = b has at least one solution for each b in Rn, then the solution is unique for each b.

Back 18

true

Front 19

If the linear transformation x --> Ax maps Rn into Rn, then A has n pivot positions.

Back 19

false, the transformation x --> Ax maps Rn to Rn on ALL square matrices, but that doesn't mean every square matrix has n pivot positions.

Front 20

If there is a b in Rn such that the equation Ax = b is inconsistent, then the transformation x --> Ax is not one-to-one.

Back 20

true

Front 21

A row replacement operation does not affect the determinant of a matrix.

Back 21

True

Front 22

The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U.

Back 22

False. If we scale any rows when getting the echelon
form, we change the determinant.

Front 23

If the columns of A are linearly dependent, then det A = 0.

Back 23

True

Front 24

det(A + B) = det A + det B

Back 24

False. detAB = (detA)(detB)

Front 25

If two row interchanged are made in succession, then the new determinant equals the old determinant.

Back 25

True

Front 26

The determinant of A is the product of the diagonal entries in A.

Back 26

False. The determinant equals the diagonal of A only when A has been reduced to triangular form.

Front 27

If det A is zero, then two rows or two columns are the same, or a row or a column is zero.

Back 27

False. ad - bc could equal 0 but that doesn't mean that two rows or two columns are the same or that a row or a column is 0.

Front 28

det A transpose = (-1) det A

Back 28

False, detA transpose = detA

Front 29

If f is a function in the vector space V of all real-valued functions on R and if f(t) = 0 for some t, then f is the zero vector in V.

Back 29

False, the 0 vector in V is the function f whose values f(t) are 0 for all t in R.

Front 30

A vector is an arrow in 3D space

Back 30

False, not every vector is an arrow.

Front 31

A subset H of a vector space V is a subspace of V if the zero vector is in H

Back 31

False, the subset has to contain the 0 vector, have closed addition, and have closed multiplication

Front 32

A subspace is also a vector space

Back 32

True

Front 33

Analogue signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter

Back 33

False, digital signals are used

Front 34

A vector is any element of a vector space.

Back 34

True

Front 35

If u is a vector in a vector space V, then (-1)u is the same as the negative of u.

Back 35

True

Front 36

A vector space is also a subspace.

Back 36

True

Front 37

R2 is a subspace of R3

Back 37

False, R2 is not even a subspace of R3

Front 38

A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v and u+v are in H, and (iii) c is a scalar and cu is in H

Back 38

True

Front 39

The null space of A is the solution set of the equation Ax = 0

Back 39

True

Front 40

The null space of an m x n matrix is in Rm

Back 40

False, it's in Rn

Front 41

The column space of A is the range of the mapping x --> Ax

Back 41

True

Front 42

If the equation Ax = b is consistent, then Col A is Rm for an m x n matrix.

Back 42

False, it has to be consistent for EVERY b, not just b

Front 43

The kernel of a linear transformation is a vector space.

Back 43

True

Front 44

Col A is the set of all vectors that can be written as Ax for some x.

Back 44

True

Front 45

A null space is a vector space

Back 45

True

Front 46

The column space of an m x n matrix is in Rm

Back 46

True

Front 47

Nul A is the kernel of the mapping
x --> Ax

Back 47

True

Front 48

The range of a linear transformation is a vector space

Back 48

True

Front 49

The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation

Back 49

True

Front 50

A single vector by itself is linearly dependent.

Back 50

False, the zero vector itself is linearly dependent.

Front 51

If H = Span{b1.....bp} then {b1.....bp} is a basis for H

Back 51

False, the set {b1.....bp} also has to be linearly independent.

Front 52

The columns of an invertible n x n matrix form a basis for Rn

Back 52

True

Front 53

A basis is a spanning set that is as large as possible

Back 53

False, it is a spanning set that is as small as possible

Front 54

In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix

Back 54

False, elementary row ops do not affect the linear dependence relations among the columns of a matrix

Front 55

A linearly independent set in a subspace H is a basis for H

Back 55

False, the set must also coincide with H

Front 56

If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V

Back 56

True

Front 57

A basis is a linearly independent set that is as large as possible

Back 57

True

Front 58

The standard method for producing a spanning set for Nul A, described in Section 4.2, sometimes fails to produce a basis for Nul A

Back 58

False, it always produces a basis

Front 59

If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A

Back 59

False, the pivot columns of A form a basis for Col A, not B

Front 60

If x is in V and if B contains n vectors, then the B-coordinate vector of x is in Rn

Back 60

True

Front 61

If PsubB is the change-of-coordinates matrix, then [x]subB = PsubBx, for x in V

Back 61

False, x = PsubB[x]subB

Front 62

The vector spaces Psub3 and R3 are isomorphic

Back 62

False, P3 is isomorphic to R4

Front 63

If B is the standard basis for Rn, then the B-coordinate vector of an x in Rn is x itself

Back 63

True

Front 64

The correspondence [x]subB --> x is called the coordinate mapping.

Back 64

False, the coordinate mapping is x --> [x]subB

Front 65

In some cases, a plane in R3 can be isomorphic to R2

Back 65

True, if the plane passes through the origin

Front 66

The number of pivot columns of a matrix equals the dimension of its column space

Back 66

True

Front 67

A plane in R3 is a 2-dimensional subspace of R3

Back 67

False, the plane must pass through the origin

Front 68

The dimension of the vector space Psub4 is 4

Back 68

False, it's 5

Front 69

If dim V = n and S is a linearly independent set in V, then S is a basis for V

Back 69

False, the set S must also have n elements

Front 70

If a set {v1....vp} spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent

Back 70

True

Front 71

R2 is a two-dimensional subspace of R3

Back 71

False, the set R2 is not even a subspace of R3

Front 72

The number of variables in the equation Ax = 0 equals the dimension of Nul A

Back 72

False, the number of free variables is equal to the dimension of Nul A

Front 73

A vector space is infinite-dimensional if it is spanned by an infinite set

Back 73

False, a basis could still have only finitely many elements, which would make the vector space finite-dimensional

Front 74

If dim V = n and if S spans V, then S is a basis for V

Back 74

False, the set S must also have n elements

Front 75

The only three-dimensional subspace of R3 is R3 itself

Back 75

True

Front 76

Col A is the set of all solutions of
Ax = b

Back 76

False, Col A is the set of all linear combinations of the columns of A