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
