I need to decide whether or not a list of subsets of $P_2$ (polynomials of degree 2 or less) are also subspaces of $P_2$:
A. $\{p(t) \mid p'(t)+1p(t)+9=0\}$
B. $\{p(t) \mid p(0)=7\}$
C. $\{p(t) \mid \int_0^1 p(t) \,\mathrm{dt}=0\}$
D. $\{p(t) \mid p'(5)=p(6)\}$.
I already know the rules:
1) check if zero vector belongs to the subset
2) check if, for scalars $a$, $b$ and vectors $v$, $w$, if $av+bw$ is a vector of the subset.
I am very confused as to how to apply the rules. I feel (according to some rough trial and error) like A, B, and D are not subspaces, while C is.
Best Answer
A,B) Not a subspace since $p(t)=0\notin S$
C) Surely, $p(t)=0\in S$. Also if $p_1(t),p_2(t)\in S$ then we can write $$\int_0^1p_1(t)dt=0\\\int_0^1p_2(t)dt=0\\\Longrightarrow\\\int_0^1ap_1(t)+bp_2(t)dt=0\\a,b\in \Bbb R\\\Longrightarrow ap_1(t)+bp_2(t)\in S$$therefore $S$ is a subspace.
D) Similarly $p(t)=0\in S$ since $p'(5)=p(6)=0$. Furthermore from $p_1(t),p_2(t)\in S$ $$p'_1(5)=p_1(6)\\p'_2(5)=p_2(6)$$and we can write $$ap_1'(5)+bp_2'(5)=ap_1(6)+bp_2(6)$$therefore $$(ap_1+bp_2)'(5)=(ap_1+bp_2)(6)$$and hence $$(ap_1+bp_2)(t)\in S$$which means that $S$ is a subset.