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Linear Algebra > True/False Chapter 4 > Flashcards

True/False Chapter 4 Flashcards

1
Q

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

A

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

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2
Q

A vector is an arrow in three-dimension space

A

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

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3
Q

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

A

false

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4
Q

A subspace is also a vector space

A

true

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5
Q

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

A

false : digital signals are used

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6
Q

A vector is any element of a vector space

A

true

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7
Q

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

A

true

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8
Q

A vector space is also a subspace

A

true

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9
Q

R2 is a subspace of R3

A

false

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10
Q

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

A

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

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11
Q

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

A

true

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12
Q

The null space of an mxn matrix is in Rm

A

false

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13
Q

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

A

true

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14
Q

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

A

false : the equation must be consistent for every b

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15
Q

The kernel of a linear transformation is a vector space

A

true

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16
Q

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

A

true

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17
Q

A null space is a vector space

A

true

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18
Q

The column space of an mxn matrix is in Rm

A

true

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19
Q

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

A

false

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20
Q

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

A

true

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21
Q

The range of a linear transformation is a vector space

A

true

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22
Q

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

A

true

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23
Q

A single vector by itself is linearly dependent

A

false : the zero vector by itself is linearly dependent

24
Q

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

A

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

25
Q

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

A

true

26
Q

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

A

false

27
Q

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

A

false

28
Q

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

A

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

29
Q

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

A

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

30
Q

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

A

true

31
Q

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

A

false

32
Q

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

A

false

33
Q

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

A

true

34
Q

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

A

false

35
Q

The vector spaces P3 and R3 are isomorphic

A

false : P3 is isomorphic to R4

36
Q

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

A

true

37
Q

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

A

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

38
Q

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

A

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

39
Q

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

A

true

40
Q

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

A

false : the plane must pass through the origin

41
Q

The dimension of the vector space P4 is 4

A

false : the dimension of Pn is n+1

42
Q

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

A

false : the set S must also have n elements

43
Q

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

A

true

44
Q

R2 is a two-dimensional subspace of R3

A

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

45
Q

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

A

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

46
Q

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

A

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

47
Q

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

A

false : the set S must also have n elements

48
Q

The only three-dimensional subspace of R3 is R3 itself

A

true

49
Q

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

A

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

50
Q

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

A

false

51
Q

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

A

true

52
Q

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

A

false

53
Q

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

A

true

54
Q

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

A

false

55
Q

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

A

false

56
Q

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

A

true

57
Q

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

A

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

Linear Algebra (10 decks)

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