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Linear Algebra > Midterm Questions From Class > Flashcards

Midterm Questions From Class Flashcards

1
Q

T or F. Every elementary row operation is reversible

A

True

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2
Q

T or F. A 5x6 matrix has six rows.

A

False

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3
Q

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

A

False

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4
Q

T or F. Two fundamental questions about a linear system involve existence and uniqueness.

A

True

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5
Q

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

A

True

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6
Q

T or F. Two matrices are row equivalent if they have the same number if rows

A

False

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7
Q

T or F. An inconsistent system has more than one solution

A

False

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8
Q

T or F. Two linear systems are equivalent if they have the same solution set

A

True

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9
Q

T or F. In some cases a matrix may be row reduced to more than one matrix in reduced echelon form using different sequences of row operations.

A

False

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10
Q

T or F. In some cases a matrix may be row reduced to more than one matrix in reduced echelon form using different sequences of row operations.

A

False

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11
Q

T or F. The row reduction algorithm applies only to augmented matrices for a linear system.

A

False

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12
Q

T or F. 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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13
Q

T or F. 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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14
Q

T or F. 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

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15
Q

T or F. The echelon form of a matrix is unique

A

False

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16
Q

T or F. The pivot position in a matrix depends on wether row interchanges are used in a row reduction process.

A

False

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17
Q

T or F. Whenever a system has free variables, the solution set contains many solutions

A

False

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18
Q

T or F. [-2;5] and [-5;2] span a line in R2

A

False

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19
Q

T or F. An example of a linear combination of vector v1 and v2 is the vector 1/2v1.

A

True

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20
Q

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

A

False

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21
Q

T or F. Asking wether the linear system corresponds to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking wether b is in Span{a1 a2 a3}

A

True

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22
Q

T or F. 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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23
Q

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

A

False

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24
Q

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

A

True

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25
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

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26
Q

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

A

True

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27
Q

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

A

False

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28
Q

T or F. The equation x=p+tv describes a line through v parallel to p

A

False

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29
Q

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

A

False

30
Q

T or F. If x is a nontrivial solution of Ax=0 then every entry in x is nonzero

A

False

31
Q

T or F. The equation Ax=b is homogeneous if the zero vector is a solution.

A

True

32
Q

T or F. The solution set of Ax=b is obtained by translating the solution set of Ax=0

A

False

33
Q

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

A

False

34
Q

T or F. If S is a linearly dependent set, then each vector is a linearly combination of the other vectors in S.

A

False

35
Q

T or F. The columns of any 4x5 matrix are linearly dependent

A

True

36
Q

T or F. If x and y are linearly independent and if {x,y,z} is linearly dependent, then z is in Spanish {x,y}.

A

True

37
Q

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

A

False

38
Q

T or F. If x and y are linearly independent, and if z is in Spanish {x,y} then {x,y,z} is linearly dependent

A

True

39
Q

How many pivot columns must a 7x5 matrix have if its columns are linearly independent? Why?

A

5

40
Q

How many pivot columns must a 5x7 matrix have if its columns span R5?

A

5

41
Q

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

A

False

42
Q

T or F. If A is an mxn matrix, then the range of the transformation x-> Axis is Rm.

A

False

43
Q

T or F. The codomain of the transformation x-> Ax is the set of all linear combinations of the columns of A

A

False (range)

44
Q

T or F. If T: Rn ->Rm is a linear transformation and if c is in Rm, then uniqueness question is “Is c in the range of T?”

A

False (existence)

45
Q

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

A

False

46
Q

T or F. AB + AC = A(B+C)

A

True

47
Q

T or F. At + Bt = (A+B)t

A

True

48
Q

T or F. (AB)C = (AC)B

A

False

49
Q

T or F. If A and B are nxn and invertible, then A^-1 B^-1 is the inverse of AB

A

False

50
Q

T or F. If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in Rn

A

True

51
Q

T or F. If A is invertible then the inverse of A^-1 is A itself

A

True

52
Q

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

A

True

53
Q

T or F. If A is an nxn matrix then the equation Ax=b has at least one solution for each b in R4

A

False

54
Q

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

A

True

55
Q

T or F. If At is not invertible then A is not invertible

A

True

56
Q

T or F. If the columns of A are linearly independent then the columns of A span Rn

A

True

57
Q

T or F. If the equation Ax=b has at least 1 solution for each b in Rn then the solution is unique for each b

A

True

58
Q

T or F. 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

A

True

59
Q

T or F. Can a square matrix with two identical columns be invertible?

A

No

60
Q

Is it possible for a 5x5 matrix to be invertible when its columns do not span R5?

A

No

61
Q

If A is invertible, can we conclude that the columns of A^-1 are linearly independent?

A

Yes

62
Q

If the equation Gx=y has more than one solution for some y in Rn can the columns of G span Rn?

A

No

63
Q

T or F. The columns of an invertible nxn matrix form a basis for Rn

A

True

64
Q

T or F. Row operations do not affect linear dependence relations among the columns of a matrix

A

True

65
Q

T or F. The null spce if an mxn matrix is a subspace of Rn

A

True

66
Q

T or F. The column space if matrix A is the set of solutions of Ax=b

A

False

67
Q

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

A

False

68
Q

T or F. Each line in Rn is a one dimensional subspace of Rn

A

False

69
Q

T or F. The dimension of Col A is the number of pivot columns

A

True

70
Q

T or F. The dimensions of Col A and Nul A add up to the number of columns of A

A

True

71
Q

T or F. If B is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in B

A

True

72
Q

T or F. The dimension of the column space of A is Rank A

A

True

Linear Algebra (2 decks)

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