Consider the subset of V={ax 2 +bx+c | a,b,c >= 0and real} with these operations
ID: 2939727 • Letter: C
Question
Consider the subset of V={ax2+bx+c | a,b,c >= 0and real} with these operations: addition is the usual addition of polynomials * (ax2+bx+c) =||ax2+||bx+||c Determine whether the structure statisfies or fails each often conditions to be a vector space. The two conditions Im having trouble with are scalarmultiplication distributes over vector addition which looks like :(x+y)=*x++y where x and y are vectors. The other one is scalar multiplication distributes over scalaraddition which looks like: (+)*x=*x+*x where x is a vector Consider the subset of V={ax2+bx+c | a,b,c >= 0and real} with these operations: addition is the usual addition of polynomials * (ax2+bx+c) =||ax2+||bx+||c Determine whether the structure statisfies or fails each often conditions to be a vector space. The two conditions Im having trouble with are scalarmultiplication distributes over vector addition which looks like :(x+y)=*x++y where x and y are vectors. The other one is scalar multiplication distributes over scalaraddition which looks like: (+)*x=*x+*x where x is a vectorExplanation / Answer
QuestionDetails: Consider the subset of V={ax2+bx+c | a,b,c >= 0and real} with these operations: addition is the usual addition of polynomials * (ax2+bx+c) =||ax2+||bx+||c Determine whether the structure statisfies or fails each often conditions to be a vector space. The two conditions Im having trouble with are scalarmultiplication distributes over vector addition which looks like :(x+y)=*x++y where x and y are vectors. The other one is scalar multiplication distributes over scalaraddition which looks like: (+)*x=*x+*x where x is a vectorI TAKE IT THAT YOU NEED A CHECK ON
1. P[V1+V2]=PV1+PV2......
AND
2.[P+Q]V1=PV1+QV1.....
WHERE P AND Q ARE SCALARS BEING ELEMENTS OF GIVENFIELD
AND V1 AND V2 ARE VECTORS BEING ELEMENTS OF V
TO CHECK
1. P[V1+V2]=PV1+PV2......
LET
V1=A1X2+B1X+C1
V2=A2X2+B2X+C2
P[V1+V2]=P[(A1+A2)X2+(B1+B2)X+(C1+C2]=|P|[(A1+A2)X2+(B1+B2)X+(C1+C2]............1
PV1=P[A1X2+B1X+C1]=|P|[A1X2+B1X+C1]
PV2=P[A2X2+B2X+C2]=|P|[A2X2+B2X+C2]
PV1+PV2=|P|[A1X2+B1X+C1]+|P|[A2X2+B2X+C2]=|P|[(A1+A2)X2+(B1+B2)X+(C1+C2]...........2
HENCE
P[V1+V2]=PV1+PV2...............OK
TO CHECK
2.[P+Q]V1=PV1+QV1....
PV1=P[A1X2+B1X+C1]=|P|[A1X2+B1X+C1]
QV1=Q[A1X2+B1X+C1]=|Q|[A1X2+B1X+C1]
PV1+QV1=|P|[A1X2+B1X+C1]+|Q|[A1X2+B1X+C1] =[|P|+|Q|][A1X2+B1X+C1].............3
[P+Q]V1=[P+Q][A1X2+B1X+C1]=[|P+Q|][A1X2+B1X+C1]...............4
EQN.3 AND EQN.4 ARE NOT EQUAL UNLESS
|P|+|Q|=|P+Q|
HENCE THIS CONDITION FAILED
SO IT IS NOT A VECTOR SPACE.