Subspace Theorem: Decide if R1 in P2

[forty]

Use the subspace Theorem to decide if the following are subspaces of P2, the vector space of all polynomials of degree at most 2.

a) R1 = {ao + a1x +a2x^2 | ao = 0}
b) R1 = {ao + a1x +a2x^2 | a1 = 1}
c) R1 = { p E P2 | p has exactly degree 2}

(for part c 'E' is 'element of')

Solutions:

a) Is a subspace, closed under vector addition and scalar multiplication

b) Isn't a subspace, vector addition doesn't hold take (a,1,c) + (d,1,e) = (a+d,2,c+e)
the value for a1 is 1 so its not a subspace.

c) Isn't a subspace, take -x^2 and x^2 under addition they equal 0 and aren't degree 2.

I'm unsure of part c whether I've just interpreted it wrongly or just made a mistake but for some reason i just doesn't feel right..

any help would be appreciated as usual, thanks :)

 

[e(ho0n3]

Everything is correct from my POV.

 

[HallsofIvy]

Use the subspace Theorem to decide if the following are subspaces of P2, the vector space of all polynomials of degree at most 2.

a) R1 = {ao + a1x +a2x^2 | ao = 0}
b) R1 = {ao + a1x +a2x^2 | a1 = 1}
c) R1 = { p E P2 | p has exactly degree 2}

(for part c 'E' is 'element of')

Solutions:

a) Is a subspace, closed under vector addition and scalar multiplication

b) Isn't a subspace, vector addition doesn't hold take (a,1,c) + (d,1,e) = (a+d,2,c+e)
the value for a1 is 1 so its not a subspace.

c) Isn't a subspace, take -x^2 and x^2 under addition they equal 0 and aren't degree 2.

Point of grammar, not mathematics: "and aren't degree 2" implies that -x^2 and x^2 are not of degree 2 which is, of course, not true. You should have said "which isn't degree 2" making it clear that it is their sum, the 0 polynomial, which is not of degree 2.

I'm unsure of part c whether I've just interpreted it wrongly or just made a mistake but for some reason i just doesn't feel right..

any help would be appreciated as usual, thanks :)

 

[forty]

Makes sense, will keep it in mind for future reference so i don't get arrested for bad grammar ;)

Love you!

 

[HallsofIvy]

You got a warning this time- next time a ticket!