True/False Chapter 4

Question 1

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

Answer 1

false : the zero vector in V is the function f for whose values f(t) are zero for all t in R

Question 2

A vector is an arrow in three-dimension space

Answer 2

false : an arrow in three-dimensional space is an example of a vector, but not every arrow is a vector

Question 3

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

Answer 3

false

Question 4

A subspace is also a vector space

Answer 4

true

Question 5

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

Answer 5

false : digital signals are used

Question 6

A vector is any element of a vector space

Answer 6

true

Question 7

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

Answer 7

true

Question 8

A vector space is also a subspace

Answer 8

true

Question 9

R2 is a subspace of R3

Answer 9

false

Question 10

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

Answer 10

false : the second and third parts of the conditions are stated incorrectly. For example, part ii) does not state that u and v represent all possible elements of H

Question 11

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

Answer 11

true

Question 12

The null space of an mxn matrix is in Rm

Answer 12

false

Question 13

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

Answer 13

true

Question 14

If the equation Ax=b is consistent, then Col A is Rm

Answer 14

false : the equation must be consistent for every b

Question 15

The kernel of a linear transformation is a vector space

Answer 15

true

Question 16

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

Answer 16

true

Question 17

A null space is a vector space

Answer 17

true

Question 18

The column space of an mxn matrix is in Rm

Answer 18

true

Question 19

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

Answer 19

false

Question 20

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

Answer 20

true

Question 21

The range of a linear transformation is a vector space

Answer 21

true

Question 22

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

Answer 22

true

Question 23

A single vector by itself is linearly dependent

Answer 23

false : the zero vector by itself is linearly dependent

Question 24

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

Answer 24

false : the set {b1,…,bp} must also be linearly independent

Question 25

The columns of an invertible nxn matrix form a basis for Rn

Answer 25

true

Question 26

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

Answer 26

false

Question 27

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

Answer 27

false

Question 28

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

Answer 28

false : the subspace spanned by the set must also coincide with H

Question 29

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

Answer 29

true : apply the spanning set theorem to V instead of H. The space V is nonzero because the spanning set uses nonzero vectors

Question 30

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

Answer 30

true

Question 31

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

Answer 31

false

Question 32

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

Answer 32

false

Question 33

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

Answer 33

true

Question 34

If Pb is the change-of-coordinates matrix, then [x]b=Pbx, for x in V

Answer 34

false

Question 35

The vector spaces P3 and R3 are isomorphic

Answer 35

false : P3 is isomorphic to R4

Question 36

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

Answer 36

true

Question 37

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

Answer 37

false : by definition, the coordinate mapping goes in the opposite direction

Question 38

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

Answer 38

true : if the plane passes through the origin, the plane is isomorphic to R2

Question 39

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

Answer 39

true

Question 40

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

Answer 40

false : the plane must pass through the origin

Question 41

The dimension of the vector space P4 is 4

Answer 41

false : the dimension of Pn is n+1

Question 42

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

Answer 42

false : the set S must also have n elements

Question 43

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

Answer 43

true

Question 44

R2 is a two-dimensional subspace of R3

Answer 44

false : the set R2 is not even a subset of R3

Question 45

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

Answer 45

false : the number of free variables is equal to the dimension of Nul A

Question 46

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

Answer 46

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

Question 47

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

Answer 47

false : the set S must also have n elements

Question 48

The only three-dimensional subspace of R3 is R3 itself

Answer 48

true

Question 49

The row space of A is the same as the column space of A^T

Answer 49

true : the rows of A are identified with the columns of A^T

Question 50

If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A

Answer 50

false

Question 51

The dimensions of the row space and column space of A are the same, even if A is not square

Answer 51

true

Question 52

The sum of the dimensions of the row space and the null space of A equals the number of rows in A

Answer 52

false

Question 53

On a computer, row operations can change the apparent rank of a matrix

Answer 53

true

Question 54

If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A

Answer 54

false

Question 55

Row operations preserve the linear dependence relations among the rows of A

Answer 55

false

Question 56

The dimension of the null space of A is the number of columns of A that are not pivot columns

Answer 56

true

Question 57

The row space of A^T is the same as the column space of A

Answer 57

true : the rows of A^T are the columns of A