Question 1

The set of solutions of the differential equation   is   . Find the general solution of the differential equation   .

Accepted Answer

A)    
B)     
C)     
D)    
E)     
A)    
B)     
C)     
D)    
E)

Question 2

Describe the zero vector (the additive identity) of the vector space   .

Accepted Answer

A)    
B)     
C)     
D)     
E)     
A)    
B)     
C)     
D)     
E)

Question 3

Write the standard basis for the vector space   .

Accepted Answer

A)    
B)     
C)     
D)    
E)     
A)    
B)     
C)     
D)    
E)

Question 4

Write the standard basis for the vector space   .

Accepted Answer

A)    
B)     
C)     
D)     
E)     
A)    
B)     
C)     
D)     
E)

Question 5

Describe the zero vector (the additive identity) of the vector space   .

Accepted Answer

A)     
B)     
C)     
D)    
E)     
A)     
B)     
C)     
D)    
E)

Question 6

The set   is linearly independent.

Accepted Answer

A) True 
 B)False

Question 7

Determine whether   is a basis for   . If it is, write   as a linear combination of the vectors in S.?

Accepted Answer

A)  S is a basis for   and   
B)  S is a basis for   and   
C)  S is a basis for   and   
D)  S is a basis for   and   
E)  S is not a basis for   
A)  S is a basis for   and   
B)  S is a basis for   and   
C)  S is a basis for   and   
D)  S is a basis for   and   
E)  S is not a basis for

Question 8

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic defined by the function below.

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 9

The set of all   lower triangular matrices is a subspace of   .

Accepted Answer

A) True 
 B)False

Question 10

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic defined by the function below.

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 11

Provided that   and   , write   as a linear combination of u and w, if possible.

Accepted Answer

A)     
B)     
C)     
D)     
E)  It is not possible to write v as a linear combination of u and w. 
A)     
B)     
C)     
D)     
E)  It is not possible to write v as a linear combination of u and w.

Question 12

Find a basis for the solution space of the following homogeneous system of linear equations.

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 13

Find the rank of the matrix

Accepted Answer

A)  2 
B)  0 
C)  4 
D)  3 
E)  1 
A)  2 
B)  0 
C)  4 
D)  3 
E)  1

Question 14

Explain why   is not a basis for   .

Accepted Answer

A)  S is linearly dependent and does not span   
B)  S is linearly dependent and spans   
C)  S is linearly independent and spans   
D)  S is linearly independent and does not span   
E)  none of these choices 
A)  S is linearly dependent and does not span   
B)  S is linearly dependent and spans   
C)  S is linearly independent and spans   
D)  S is linearly independent and does not span   
E)  none of these choices

Question 15

The set of solutions of the differential equation   is   . Find the general solution of the differential equation   .

Accepted Answer

A)    
B)     
C)    
D)     
E)    
A)    
B)     
C)    
D)     
E)

Question 16

The set of all functions such that   is a subspace of   .

Accepted Answer

A) True 
 B)False

Question 17

Determine whether the set of all   diagonal matrices with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

Accepted Answer

A)  This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property 
B)  This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity 
C)  This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication 
D)  This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity 
E)  This set is a vector space. All ten vector space axioms hold. 
A)  This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property 
B)  This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity 
C)  This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication 
D)  This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity 
E)  This set is a vector space. All ten vector space axioms hold.

Question 18

Explain why   is not a basis for   .

Accepted Answer

A)  S is linearly independent and spans   
B)  S is linearly dependent and spans   
C)  S is linearly dependent and does not span   
D)  S is linearly independent and does not span   
E)  none of these choices 
A)  S is linearly independent and spans   
B)  S is linearly dependent and spans   
C)  S is linearly dependent and does not span   
D)  S is linearly independent and does not span   
E)  none of these choices

Question 19

Explain why   is not a basis for   .

Accepted Answer

A)  S is linearly independent and spans   
B)  S is linearly dependent and spans   
C)  S is linearly independent and does not span   
D)  S is linearly dependent and does not span   
E)  none of these choices 
A)  S is linearly independent and spans   
B)  S is linearly dependent and spans   
C)  S is linearly independent and does not span   
D)  S is linearly dependent and does not span   
E)  none of these choices

Question 20

Determine whether B is in the column space of A. If it is, write B as a linear combination of the column vectors of A..

Accepted Answer

A)    is in the column space of A and   
B)    is in the column space of A and   
C)    is in the column space of A and   
D)    does not belong to the column space of   
E)  none of these choices 
A)    is in the column space of A and   
B)    is in the column space of A and   
C)    is in the column space of A and   
D)    does not belong to the column space of   
E)  none of these choices

The set of solutions of the differential equation is . Find the general solution of the differential equation .

Describe the zero vector (the additive identity) of the vector space .

Write the standard basis for the vector space .

Write the standard basis for the vector space .

Describe the zero vector (the additive identity) of the vector space .

The set is linearly independent.

Determine whether is a basis for . If it is, write as a linear combination of the vectors in S.?

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic defined by the function below.

The set of all lower triangular matrices is a subspace of .

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic defined by the function below.

Provided that and , write as a linear combination of u and w, if possible.

Find a basis for the solution space of the following homogeneous system of linear equations.

Find the rank of the matrix

Explain why is not a basis for .

The set of solutions of the differential equation is . Find the general solution of the differential equation .

The set of all functions such that is a subspace of .

Determine whether the set of all diagonal matrices with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

Explain why is not a basis for .

Explain why is not a basis for .

Determine whether B is in the column space of A. If it is, write B as a linear combination of the column vectors of A..

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Exam 4: Vector Spaces

The set of solutions of the differential equation is . Find the general solution of the differential equation .

Describe the zero vector (the additive identity) of the vector space .

Write the standard basis for the vector space .

Write the standard basis for the vector space .

Describe the zero vector (the additive identity) of the vector space .

The set is linearly independent.

Determine whether is a basis for . If it is, write as a linear combination of the vectors in S.?

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic defined by the function below.

The set of all lower triangular matrices is a subspace of .

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic defined by the function below.

Provided that and , write as a linear combination of u and w, if possible.

Find a basis for the solution space of the following homogeneous system of linear equations.

Find the rank of the matrix

Explain why is not a basis for .

The set of solutions of the differential equation is . Find the general solution of the differential equation .

The set of all functions such that is a subspace of .

Determine whether the set of all diagonal matrices with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

Explain why is not a basis for .

Explain why is not a basis for .

Determine whether B is in the column space of A. If it is, write B as a linear combination of the column vectors of A..

Linear Equations

Matrices

Determinants

Inner Product Spaces

Linear Transformations

Eigenvalues Eigenvectors

Filters