Question

Let H be the set of all points of the form ​(s,sminus​1). Determine whether H is a vector space. If it is not a vector​ space, determine which of the following properties it fails to satisfy. 1. Contains a zero vector 2. Closed under vector addition 3. Closed under multiplication by scalars A. H is not a vector space because it does not contain a zero vector. B. H is a vector space. C. H is not a vector space because fails to satisfy all three properties. D. H is not a vector space because it is not closed under vector addition.

Answer 1

Answer:

Question Details:

Solve the problem.

Let H be the set of all points of the form (s, s-1). Determinewhether H is a vector space. If it is not a vector space, determinewhich of the following properties it fails to satisfy.

LET ANY 2 VECTORS IN H BE U=[A,A-1] AND V=[B,B-1]

LET K BE ANY SCALAR

A: Contains zero vector

LET THE ZERO VECTOR BE [Z,Z-1].

THEN WE SHOULD HAVE

A+Z=A ....THAT IS

[A+Z,A+Z-2]=[A,A-1]

HENCE A=A+Z....Z=0

A+Z-2=A-1....Z=1...CONTRADICTORY

SO NO ZERO ELEMENT EXISTS.

B: Closed under vector addition

A+B=[A,A-1]+[B,B-1]=[A+B,A+B-2]....NOT AN ELEMENT OF H AS IICOORDINATE SHOULD BE A+B-1.

SO NOT CLOSED IN ADDITION.

C: Closed under multiplication by scalars

K[A,A-1] =[KA,KA-K].......NOT AN ELEMENT OF H AS II COORDINATESHOULD BE KA-1.

SO NOT CLOSED IN SCALAR MULTIPLICATION.

Student Response

Percent              Correct                  Student                 Answer

Value                Response  Response               Choices

 

0.0%                                                             H is not a vector space;

                                                                                 does not                                      

                                                                                 contain zero vector

             

0.0%                                                                       b.H is a vector space.

                                     

0.0%                                                                 c. H is not a vector space;  

                                                                                   not closed under    

                                                                                      vector addition

                                                                                 

 

100.0%                                                             d. H is not a vector  space

                                                           ; fails to satisfy all three  properties

                                                                         

                                 

                                                             

AS YOU CAN SEE ABOVE H IS NOT A VECTOR SPACE .

IT FAILED TO SATISFY THE ABOVE 3 PROPERTIES.

PLEASE NOTE THAT FAILURE TO SATISFY EVEN A SINGLE PROPERTY

IS ENOUGH TO RULE IT OUT AS A VECTOR SPACE