I am wondering if someone could give me a bump in the right direction, I am not sure if my answers are right. Also, you know of any tips for the future when dealing with these types of questions.
For any non-negative integer k, let C^k(R) denote the real vector space of all continuous functions f:R→R such that the derivatives f′,f′′,...,f^(k) exist and are continuous.
(a) Show that the set U of all functions f ∈ C^2(R) such that f′′ + f′ + f = 0 (1) forms a subspace.
U is non-empty because there exists a zero function (f=0) that satisfies equation (1)
Let u and v be in U, (u+v)''+(u+v)'+(u+v) = (u''+u'+u)+(v''+v'+v)=0+0=0 thus showing U is closed under addition
let c be a scalar (cf)''+(cf)'+(cf)=c(f''+f'+f)=c0=0 thus showing U is closed under scalar multiplication
(b) Prove that the list 1, x, |x| is linearly independent in C^0(R).
In C^0(R), f=0 c1+cx+c|x|=0 thus c=0 to satisfy this equation. Proving they are linearly independent.
Part $a$ seems ok.
For part $b$,
Suppose $$c_1 + c_2x+c_3|x|=0$$
If $x=0$, you should be able to evaluate the value of $c_1$.
$\forall x>0$, then $$c_1+(c_2+c_3)x=0$$
In particular, you can let $x=1$.
$\forall x <0$, $$c_1 +(c_2-c_3)x=0$$
In particular you let let $x=-1$.
Try solving to solve for $c_1, c_2, c_3$.