求教0568 第22题

又是一天UID: 2604890

Let C(R)be the collection of all continuous functions from (R) to (R). Then (R)is a real vector space with
pointwise addition and scalar multiplication defined by
( f + g)(x)= f (x) + g(x) and (rf (x) )= rf (x)
for all f , g 属于C（R） and all r, x 属于（R）. Which of the following are subspaces of C（R） ?
I. 􀁞 f : f is twice differentiable and f "(x)- 2 f '(x) + 3 f (x) = 0 for all x 􀁠
II. 􀁞g : g is twice differentiable and g''(x) = 3g'(x) for all x 􀁠
III. 􀁞h : h is twice differentiable and h''(x) + h(x) + 1 for all x 􀁠
(A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III

zhunshangUID: 2556536

A subspace of a vector space is closed under addition and scalar multiplication.
Take I for example, if f and g belong to the set I and r is a real number, you can easily check that f+g, rf are also in I.  Similarly, II is a subspace of C(R).
As for III, suppose f,g in III, then (f+g)''(x)=f''(x)+g''(x)=f(x)+g(x)+2!=(f+g)(x)+1, which means III is not closed under addition.
So I think the answer is B. Good luck.

cobyssUID: 2638623

Wow, I just met this this fellow yesterday, not in 05 though.

Ah, the third one is not closed under the multiplicative operation hence CDE should be ruled out. Now we just need to test the second one. You will definitely see that it's correct too after pluging in f+g and rg to take derivatives. Therefore, the first two is both true and the answer should be B.

In fact I'm gonna take GRE Sub tomorrow too. Good luck!