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

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