Practice Problems

Question 1

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

Answer 1

False

Question 2

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

Answer 2

False

Question 3

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

Answer 3

True

Question 4

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

Answer 4

True

Question 5

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

Answer 5

False

Question 6

The Echelon form of a matrix is unique

Answer 6

False

Question 7

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

Answer 7

False

Question 8

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

Answer 8

True

Question 9

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

Answer 9

False. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty.

Question 10

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

Answer 10

True

Question 11

Suppose a 3 x 5 coefficient matrix for a system has three pivot columns is the system consistent? Wire why not?

Answer 11

Yes, the system is consistent because with three pivots, there must be a pivot in the third bottom row of the coefficient matrix the reduced echelon form cannot contain a row of the form [ 0 0 0 0 0 1]

Question 12

Suppose a system of linear equations has a 3 x 5 augmented matrix whose fifth column is a pivot column. Is the system consistent? Wire why not

Answer 12

The system is inconsistent because the pivot in the fifth column means that there is a row of [ 0 0 0 0 01] in the reduced echelon form. Since the matrix is the augmented matrix for our system, then this is an evil row.

Question 13

Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.

Answer 13

If the coefficient matrix has a pivot position in every row, then there is a pivot position in the bottom row, and there is no room for fit in the augmented column. So the system is consistent by theorem two

Question 14

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column explain why the system has a unique solution

Answer 14

Since there are three pivots one in each row, the augmented matrix must reduce no matter what the values of ABC the solution exists and is unique.

Question 15

Restate the last sentence in theorem two using the concept of pivot columns.” if a linear system is consistent, then the solution is unique if and only if_____”

Answer 15

Every column in the coefficient matrix is a pivot column otherwise there are infinitely many solutions

Question 16

What would you have to know about the pivot columns in any augment matrix in order to know that the linear system is consistent and has a unique solution

Answer 16

Every column in the augmented matrix except the right most column is a pivot column, and the right most column is not a pivot column.

Question 17

A system of linear equations with fewer equations than known is sometimes called an undetermined system. Suppose that such a system happens to be consistent. Explain why there must be an infinite number of solutions.

Answer 17

An undetermined system always has more variable equations. There cannot be more basic variables than there are equations. There must be at least one free variable. Such a variable may be assigned, infinitely many different values. If the system is consistent, each different values of a free variable will produce a different solution.

Question 18

A system of linear equations with more equations than unknown is sometimes called an overdetermined system. Can such a system be consistent?

Answer 18

Yes, our system of linear equations with more equations than unknown can be consistent

Question 19

Another notation for the vector [-4/3] is [-4 3]

Answer 19

False, the alternative notation for a column vector is (-4, 3) using parentheses and commas

Question 20

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

Answer 20

False. Plot the points to verify this.

Question 21

An example of a linear combination of vectors V1 and V2 is the vector 1/2 V1

Answer 21

True

Question 22

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

Answer 22

True

Question 23

The set span{u, v} It’s always visualized as a plane through the origin.

Answer 23

False. The statement is often true, but the span can also be a line or the zero vector.

Question 24

Any list of five real numbers as a vector in R5

Answer 24

True

Question 25

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

Answer 25

True

Question 26

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

Answer 26

False

Question 27

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

Answer 27

True

Question 28

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

Answer 28

True

Question 29

The equation AX = B is referred to as a vector equation

Answer 29

False, that is the matrix equation

Question 30

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

Answer 30

True

Question 31

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

Answer 31

False

Question 32

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

Answer 32

True

Question 33

If the columns of a nxm matrix A span Rm then the equation Ax= b is consistent first each b in R^m

Answer 33

True

Question 34

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

Answer 34

True

Question 35

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

Answer 35

True

Question 36

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

Answer 36

True

Question 37

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]

Answer 37

True

Question 38

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

Answer 38

True

Question 39

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

Answer 39

False

Question 40

In A is an mxn matrix whose columns do not span Rm then the equation Ax=b is inconsistent for some b in R^m

Answer 40

True

Question 41

Let A be a 3x2 matrix. Explain why the equation Ax=b cannot be consistent for all b in R^3. Generalize your argument to the case of an arbitrary A with more rows than columns.

Answer 41

A 3x2 matrix has three rows and two columns. With only two columns, A can have at most two pivot columns and so A has at most two pivot positions, which is not enough to fill all three rows. By Theorem 4, the equation Ax=b cannot be consistent for all b in R^3. Generally, if A is an mxn matrix m>n then A can have at most n pivot positions which is not enough to fill all m rows. Thus the equation Ax=b cannot be consistent for all b in R^3

Question 42

Could a set of three vectors in R^4 span all of R^4? Explain. What about n vectors in R^m when n is less than m?

Answer 42

A set of three vectors in cannot span R^4. Reason: the matrix A whose columns are these three vectors has four rows. To have a pivot in each row, A would have to have at least 4 columns which is not the case. Since A does not have a pivot in every row, its columns do not span R^4 by theorem 4.

Question 43

A homogeneous equation is always consistent

Answer 43

True

Question 44

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

Answer 44

False. It gives an implicit description of

Question 45

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

Answer 45

False

Question 46

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

Answer 46

False

Question 47

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

Answer 47

False

Question 48

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

Answer 48

False

Question 49

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

Answer 49

False

Question 50

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

Answer 50

False

Question 51

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

Answer 51

False

Question 52

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

Answer 52

False

Question 53

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

Answer 53

False

Question 54

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

Answer 54

False

Question 55

The columns of any 4x5 matrix are linearly dependent

Answer 55

True

Question 56

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

Answer 56

True

Question 57

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

Answer 57

True

Question 58

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

Answer 58

False

Question 59

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

Answer 59

True

Question 60

If a set in R^n is linearly dependent then the set contains more vectors than there are entries in each vector

Answer 60

False

Question 61

A linear transformation is a special type of function

Answer 61

True

Question 62

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

Answer 62

False

Question 63

If A is an mxn matrix then the range of the transformation x-> Ax is R^m

Answer 63

False

Question 64

Every linear transformation is a matrix transformation

Answer 64

False

Question 65

A transformation T is linear if and only if T(c1v1 + c2v2) = c1T1(v1) + c2T(v2). For all v1 and v2 in the domain of T and for all scalars c1 and c2

Answer 65

True

Question 66

Every matrix transformation x-> Axis is the set of all linear combinations of the columns of A

Answer 66

True

Question 67

The codomain of the transformation x-> Axis is the set of all linear combinations if the columns of A

Answer 67

False

Question 68

If T: R^n -> R^m is a linear transformation and if c is in R^m, then a uniqueness question is “Is c in the range of T?”

Answer 68

False

Question 69

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

Answer 69

True

Question 70

The superposition principle is a physical description of a linear transformation

Answer 70

True

Question 71

A linear transformation T: R^n -> R^m is completely determined by its effect on the columns of the nxn identity matrix

Answer 71

True

Question 72

If T: R^2 -> R^2 rotates vectors about the origin through the angle y, then T is a linear transformation

Answer 72

True

Question 73

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

Answer 73

False

Question 74

A mapping T: R^n -> R^m is onto R^m if every vector x in R^n maps onto some vector in R^m

Answer 74

False

Question 75

If A is a 3x2 matrix then the transformation x -> Axis can not be one-one

Answer 75

False

Question 76

Not every linear transformation from R^n to R^m is a matrix transformation.

Answer 76

False

Question 77

The columns of the standard matrix for a linear transformation from R^n to R^m are the images of the columns of the nxn identity matrix

Answer 77

True

Question 78

The standard matrix of a linear transformation from R^2 to R^2 that reflects points through a horizontal axis the vertical axis or the origin has the form ( a 0, 0 d)

Answer 78

True

Question 79

A mapping T: R^n -> R^n is one to one if each vector in R^n maps onto a unique vector in R^m

Answer 79

False

Question 80

If A is a 3x2 matrix then the transformation x-> Axis can not map R^2 onto R^3

Answer 80

True

Question 81

The determinant of A is the product of the diagonal entries in A

Answer 81

False

Question 82

An elementary row operation on A does not change the determinant

Answer 82

False

Question 83

Det(A) det(B) = detAB

Answer 83

True

Question 84

If lambda + 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A

Answer 84

False

Question 85

If A is 3x3 with columns a1 a2 and a3 then det A equals the volume of the parallelepiped determined by a1 a2 and a3

Answer 85

False

Question 86

Det(A^T) = (-1) det(A)

Answer 86

False

Question 87

The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A

Answer 87

True

Question 88

The row replacement operation on A does not change the eigenvalue

Answer 88

False

Question 89

If Ax= lamdba x for some vector x then lambda is an eigenvalue of A.

Answer 89

False. It must have a no trivial solution

Question 90

A matrix A is nit invertible if and only if 0 is an eigenvalue of A

Answer 90

True

Question 91

The number c is an eigenvalue of A if and only if 0 is an eigenvalue of A

Answer 91

True

Question 92

A number c is an eigenvalue of A if and only if the equation (A-cI)c =0 has a nontrivial solution

Answer 92

True

Question 93

Finding an eigenvector of A may be difficult but checking whether a given vector is in fact an eigenvector is easy

Answer 93

True

Question 94

To find eigenvalues of A reduce A to echelon form

Answer 94

False

Question 95

If Ax= lambda x for some scalar lmbda then x is an eigenvector of A

Answer 95

False

Question 96

If v1 and v2 are linearly independent eigenvectors then they correspond to distinct eigenvalues

Answer 96

False

Question 97

A steady state vector if a stochastic matrix is actually an eigenvector

Answer 97

True

Question 98

The eigenvalue of a matrix are on its main diagonal

Answer 98

False

Question 99

An eigenspace of A is a null space of a certain matrix

Answer 99

True