Midterm Questions From Class

Question 1

T or F. Every elementary row operation is reversible

Answer 1

True

Question 2

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

Answer 2

False

Question 3

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

Answer 3

False

Question 4

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

Answer 4

True

Question 5

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

Answer 5

True

Question 6

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

Answer 6

False

Question 7

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

Answer 7

False

Question 8

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

Answer 8

True

Question 9

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.

Answer 9

False

Question 10

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.

Answer 10

False

Question 11

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

Answer 11

False

Question 12

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

Answer 12

True

Question 13

T or F. Finding a parametric description of the solution set of a linear system is the same as solving the system.

Answer 13

True

Question 14

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

Answer 14

False

Question 15

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

Answer 15

False

Question 16

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

Answer 16

False

Question 17

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

Answer 17

False

Question 18

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

Answer 18

False

Question 19

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

Answer 19

True

Question 20

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

Answer 20

False

Question 21

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}

Answer 21

True

Question 22

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.

Answer 22

True

Question 23

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

Answer 23

False

Question 24

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

Answer 24

True

Question 25

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.

Answer 25

True

Question 26

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

Answer 26

True

Question 27

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

Answer 27

False

Question 28

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

Answer 28

False

Question 29

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

Answer 29

False

Question 30

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

Answer 30

False

Question 31

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

Answer 31

True

Question 32

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

Answer 32

False

Question 33

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

Answer 33

False

Question 34

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

Answer 34

False

Question 35

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

Answer 35

True

Question 36

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}.

Answer 36

True

Question 37

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

Answer 37

False

Question 38

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

Answer 38

True

Question 39

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

Answer 39

5

Question 40

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

Answer 40

5

Question 41

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

Answer 41

False

Question 42

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

Answer 42

False

Question 43

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

Answer 43

False (range)

Question 44

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?”

Answer 44

False (existence)

Question 45

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

Answer 45

False

Question 46

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

Answer 46

True

Question 47

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

Answer 47

True

Question 48

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

Answer 48

False

Question 49

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

Answer 49

False

Question 50

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

Answer 50

True

Question 51

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

Answer 51

True

Question 52

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

Answer 52

True

Question 53

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

Answer 53

False

Question 54

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

Answer 54

True

Question 55

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

Answer 55

True

Question 56

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

Answer 56

True

Question 57

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

Answer 57

True

Question 58

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

Answer 58

True

Question 59

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

Answer 59

No

Question 60

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

Answer 60

No

Question 61

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

Answer 61

Yes

Question 62

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

Answer 62

No

Question 63

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

Answer 63

True

Question 64

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

Answer 64

True

Question 65

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

Answer 65

True

Question 66

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

Answer 66

False

Question 67

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

Answer 67

False

Question 68

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

Answer 68

False

Question 69

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

Answer 69

True

Question 70

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

Answer 70

True

Question 71

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

Answer 71

True

Question 72

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

Answer 72

True