TheWrathOfMath
Junior Member
- Joined
- Mar 31, 2022
- Messages
- 162
V=P2[x]
U={p(x)∈V | a+b+c=0}
W={p(x)∈V |p(0)=p(1)}
a) Is U+W=V ?
b) Is U⊕W=V ?
Is what I did so far correct for (a), and if so, how do I proceed?
U={p(x)∈V | a+b+c=0}
=>
U={a+bx+cx^2 | a+b+c=0}
=>
U={a+bx+(-a-b)x^2 | a,b ∈ F}
W={p(x)∈V |p(0)=p(1)}
=>
W={a+bx+cx^2 | a+b(0)+c(0)^2 = a+b(1)+c(1)^2}
=>
W={a+bx+cx^2 | a=a+b+c}
=>
W={a+bx+cx^2 | c= -b}
=>
W={a+bx+(-b)x^2 | a,b ∈ F}
U+W = a+bx+(-a-b)x^2 + a+bx+(-b)x^2
= 2a+2bx+(-a-2b)x^2
.
.
.
How do I proceed from here?
I know that P2(x) is of the form: a+bx+cx^2.
U={p(x)∈V | a+b+c=0}
W={p(x)∈V |p(0)=p(1)}
a) Is U+W=V ?
b) Is U⊕W=V ?
Is what I did so far correct for (a), and if so, how do I proceed?
U={p(x)∈V | a+b+c=0}
=>
U={a+bx+cx^2 | a+b+c=0}
=>
U={a+bx+(-a-b)x^2 | a,b ∈ F}
W={p(x)∈V |p(0)=p(1)}
=>
W={a+bx+cx^2 | a+b(0)+c(0)^2 = a+b(1)+c(1)^2}
=>
W={a+bx+cx^2 | a=a+b+c}
=>
W={a+bx+cx^2 | c= -b}
=>
W={a+bx+(-b)x^2 | a,b ∈ F}
U+W = a+bx+(-a-b)x^2 + a+bx+(-b)x^2
= 2a+2bx+(-a-2b)x^2
.
.
.
How do I proceed from here?
I know that P2(x) is of the form: a+bx+cx^2.