Question 1

The set  spans  .

Accepted Answer

A) True 
 B)False

Question 2

Determine whether the set of all fourth-degree polynomial functions as given below, whose graphs pass through the origin 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.
Additive identity
Additive inverse
Associative property
Scalar identity 
D)  This set is a vector space. All ten vector space axioms hold. 
E)  This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 3

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

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 4

Write the standard basis for the vector space  .

Accepted Answer

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)

Question 6

Find the coordinate matrix of  relative to the standard basis in  .

Accepted Answer

A)   
B)   
C)    
D)   
E)

Question 7

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)

Question 8

Identify the graph of the function  .

Accepted Answer

A)  The graph of this equation is a hyperbola centered at  with transverse axis parallel to the x-axis. 
B)  The graph of this equation is a hyperbola centered at  with transverse axis parallel to the y-axis. 
C)  The graph of this equation is an ellipse centered at  with major axis parallel to the x-axis. 
D)  The graph of this equation is an ellipse centered at  with major axis parallel to the x-axis. 
E)  The graph of this equation is an ellipse centered at  with major axis parallel to the y-axis.

Question 9

the following subset of  is a subspace of  .The set of all constant functions:

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)

Question 11

The set  spans  .

Accepted Answer

A) True 
 B)False

Question 12

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

Accepted Answer

A)    
B)   
C)    
D)    
E)    ?

Question 13

Write the standard basis for the vector space  .

Accepted Answer

A)   
B)    
C)    
D)    
E)

Question 14

The set  spans  .

Accepted Answer

A) True 
 B)False

Question 15

The set  is a basis for  .

Accepted Answer

A) True 
 B)False

Question 16

Determine whether the set of all second-degree polynomial functions as given below, whose graphs pass through the origin 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.
Additive identity
Additive inverse
Associative property
Scalar identity 
D)  This set is a vector space. All ten vector space axioms hold. 
E)  This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 17

Given that  , the coordinate matrix of x relative to a (nonstandard) basis  . Find the coordinate vector of x relative to the standard basis in  .

Accepted Answer

A)   
B)    
C)    
D)    
E)

Question 18

Find a basis for the column space of the matrix  .

Accepted Answer

A)   
B)    
C)    
D)    
E)

Question 19

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

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 20

Sketch the graph of the equation  .

Accepted Answer

A)    
B)    
C)    
D)    
E)

Exam 4: Vector Spaces

The set spans .

Determine whether the set of all fourth-degree polynomial functions as given below, whose graphs pass through the origin with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

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

Write the standard basis for the vector space .

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

Find the coordinate matrix of relative to the standard basis in .

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

Identify the graph of the function .

the following subset of is a subspace of .The set of all constant functions:

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

The set spans .

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

Write the standard basis for the vector space .

The set spans .

The set is a basis for .

Determine whether the set of all second-degree polynomial functions as given below, whose graphs pass through the origin with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

Given that , the coordinate matrix of x relative to a (nonstandard) basis . Find the coordinate vector of x relative to the standard basis in .

Find a basis for the column space of the matrix .

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

Sketch the graph of the equation .