Which of the following subsets of P2 are subspaces P2 pt int

Which of the following subsets of P_2 are subspaces P_2? {p(t)| integral^1_0 p(t)dt = 0} {p(t) |p\'(t) + 3p(t) + 2 = 0} {p(t) | p(2) = 0} {p(t) |p(6) = 6} {p(t) | p\'(t) is constant} {p(t) |p(-t) = p(t) for all t}

Solution

E) p\'(t) = constant
That constant could be zero, and it could be integrated to give another zero constant. It includes the zero vector.

(F) p(-t) = -p(t)
This is the definition of odd functions generally, but a special \'exception\' exists:
p(t) = 0
p(-t) = 0 = -p(t)
This includes the zero vector.

(D) p(6) = 6
3 can have scalar multiplication and addition applied to it, but it doesn\'t include the zero vector.

(A) It is zero.
(C) Again, zero.

(6) The differential equation is zero, but the p function isn\'t directly.
If you make p(t) = 0, you get:
p\'(t) + 3p(t) + 2 = 0
( 0 ) + 3( 0 ) + 2 = 0
2 = 0
This is not true. The zero vector is not included.

Thus: A,C , E and F