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Linear Algebra > True/False Chapter 1 > Flashcards

True/False Chapter 1 Flashcards

1
Q

Every elementary row operation is reversible

A

true

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2
Q

A 5x6 matrix has 6 rows

A

false : 5 rows and 6 columns

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3
Q

The solution set of a linear system involving variables x1,…,xn is a list of numbers (s1,…,sn) that makes each equation in the system a true statement when the values s1,…sn are substituted for x1,…,xn, respectively

A

false : the description given applies to a single solution. The solution set consists of all possible solutions. Only in special cases does the solution set consist of exactly one solution

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4
Q

Two fundamental questions about a linear system involve uniqueness and existence

A

true

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5
Q

Elementary row operations on an augmented matrix never change the solution set of the associated linear system

A

true

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6
Q

Two matrices are row equivalent if they have the same number of rows

A

false: the definition of row equivalent requires that there exist a sequence of row operations that transform one matrix into the other

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7
Q

An inconsistent system has more than one solution

A

false : by definition, an inconsistent system has no solution

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8
Q

Two linear systems are equivalent if they have the same solution set

A

true

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9
Q

In some cases, a matrix can be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations

A

false : each matrix is row equivalent to one and only one reduced echelon matrix

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10
Q

The row reduction algorithm applies only to augmented matrices for a linear system

A

false

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11
Q

A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix

A

true

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12
Q

Finding a parametric description of the solution set of a linear system is the same as solving the system

A

true

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13
Q

If one row in an echelon form of an augmented matrix is [0 0 0 5 0] , then the associated linear system is inconsistent

A

false : might or might not be consistent

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14
Q

The echelon form of a matrix is unique

A

false : only the reduced form is unique

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15
Q

The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process

A

false : the pivot positions are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix

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16
Q

Reducing a matrix to echelon form is called the forward phase of the row reduction process

A

true

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17
Q

Whenever a system has free variables, the solution set contains many solutions

A

false: the existence of at least one free solution is not related to the presence or absence of free variables

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18
Q

A general solution of a system is an explicit description of all solutions of the system

A

true

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19
Q

The point in the plane corresponding to (-2, 5) and (-5, 2) lie on a line through the origin

A

false : if (-5, 2) were on a line through (-2, 5) and the origin, then (-5, 2) would have to be a multiple of (-2, 5), which is not the case

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20
Q

An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1

A

true

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21
Q

The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b

A

true

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22
Q

The set Span {u, v} is always visualized as a plane through the origin

A

false : the statement is often true, but Span {u, v} is not a plane when u is a multiple of v, or when u is the zero vector

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23
Q

Any list of five real numbers is a vector in R5

A

true

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24
Q

The vector u results when a vector u-v is added to the vector v

A

true

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25
Q

The weights c1,…cp in a linear combination c1v1 + … + cpvp cannot all be zero

A

false

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26
Q

When u and v are nonzero vectors, Span{u, v} contains the line through u and the origin

A

true

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27
Q

Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span{a1, a2, a3}

A

true

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28
Q

The equation Ax=b is referred to as a vector equation

A

false : it is a matrix equation

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29
Q

A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution

A

true

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30
Q

The equation Ax=b is consistent if the augmented matrix [A b] has a pivot in every row

A

false

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31
Q

The first entry in the product Ax is a sum of products

A

true

32
Q

If the columns of an mxn matrix A span Rm, then the equation Ax=b is consistent for each b in Rm

A

true

33
Q

If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row

A

true

34
Q

Every matrix equation Ax=b corresponds to a vector equation with the same solution set

A

true

35
Q

Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x

A

true

36
Q

the solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax = b, if A = [a1 a2 a3]

A

true

37
Q

If the equation Ax=b is inconsistent, then b is not in the set spanned by the columns of A

A

true : saying that b is not in the set spanned by the columns of A is the same as saying that b is not a linear combination of the columns of A

38
Q

If the augmented matrix [A b] has a pivot position in every row, then the equation Ax=b is inconsistent

A

false

39
Q

If A is an mxn matrix whose columns do not span Rm, then the equation Ax=b is inconsistent for some b in Rm

A

true

40
Q

A homogeneous equation is always consistent

A

true

41
Q

The equation Ax=0 gives an explicit description of its solution set

A

false : the equation Ax=0 gives an implicit description of its solution set

42
Q

The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable

A

false : the equation Ax=0 always has the trivial solution

43
Q

The equation x = p + tv describes a line through v parallel to p

A

false : the line goes through p parallel to v

44
Q

The solution set of Ax=b is the set of all vectors of the form w = p + vn, where vn is any solution of the equation Ax=0

A

false : the solution set could be empty, the statement is true only when there exists a vector p such that Ap=b

45
Q

If x is a nontrivial solution of Ax=0, then every entry of x is nonzero

A

false : A nontrivial solution of Ax=0 is any nonzero x that satisfies the equation

46
Q

The equation x=x2u+x3v, with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin

A

true

47
Q

The equation Ax=b is homogeneous is the zero vector is a solution

A

true : if the zero vector is a solution, then b=Ax=A0=0

48
Q

The effect of adding p to a vector is to move the vector in a direction parallel to p

A

true

49
Q

The solution set of Ax=b is obtained by translating the solution set of Ax=0

A

false : the statement is true only when the solution set of Ax=0 is nonempty

50
Q

The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution

A

false : a homogeneous system always has the trivial solution

51
Q

If S is a linearly independent set, then each vector is a linear combination of the other vectors in S

A

false

52
Q

The columns of any 4x5 matrix are linearly dependent

A

true

53
Q

If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span{x, y}

A

true

54
Q

Two vectors are linearly dependent if and only if they lie on a line through the origin

A

true

55
Q

If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent

A

false

56
Q

If x and y are linearly independent, and if z is in Span{x, y}, then {x, y, z} is linearly dependent

A

true

57
Q

If a set in Rn is linearly dependent, then the set contains more vectors than there are entries in each vector

A

false

58
Q

A linear transformation is a special type of function

A

true

59
Q

If A is a 3x5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is R3

A

false : the domain is R5

60
Q

If A is an mxn matrix, then the range of the transformation x–> Ax is Rm

A

false : the range in the set of all linear combinations of the columns of A

61
Q

Every linear transformation is a matrix transformation

A

false

62
Q

A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2

A

true

63
Q

Every matrix transformation is a linear transformation

A

true

64
Q

The codomain of the transformation x–>Ax is the set of all linear combinations of the columns of A

A

false : if A is an mxn matrix, the codomain is Rm

65
Q

If T:Rn–>Rm is a linear transformation and if c (vector) is in Rm, then a uniqueness question is “Is c in the range of T?”

A

false : the question is an existence question

66
Q

A linear transformation preserves the operations of vector addition and scalar multiplication

A

true

67
Q

The superposition principle is a physical description of a linear transformation

A

true

68
Q

A linear transformation T:Rn–>Rm is completely determined by its effect on the columns of the nxn identity matrix

A

true

69
Q

If T:R2–>R2 rotates vectors about the origin through an angle §, then T is a linear transformation

A

true

70
Q

When two linear transformations are performed one after another, the combined effect may not always be a linear transformation

A

false

71
Q

A mapping T:Rn–>Rm is onto Rm if every vector x in Rn maps onto some vector in Rm

A

false : any function from Rn to Rm maps each vector onto another vector

72
Q

If A is a 3x2 matrix, then the transformation x–>Ax cannot be one-to-one

A

false

73
Q

Not every linear transformation from Rn to Rm is a matrix transformation

A

false

74
Q

The columns of the standard matrix from a linear transformation from Rn to Rm are the images of the columns of the nxn identity matrix

A

true

75
Q

The standard matrix of a linear transformation from R2 to R2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form a 0
0 d
where a and d are + or - 1

A

true

76
Q

A mapping T : Rn–>Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm

A

false : any function from Rn to Rm maps a vector onto a single (unique) vector

77
Q

If A is a 3x2 matrix, then the transformation x–>Ax cannot map R2 onto R3

A

true

Linear Algebra (10 decks)

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