Let V C ( a, b ) be the vector space consisting of all functions f(t) which are defined and continuous on the interval 0 curren t curren 1 Which of the following conditions define subspaces of V Explain your answer (a) f(0) 0 (b) f(0) 2 (l) (c) f(0)f(l) 1 (d) f(0) 0 or f(l) 0 (e) f(I r) tf(t) (f) f(1 t) l f(t) (g) (h) (i) f(1) I f(t)dt (t 1)f(r)dt 0 f (s) sin s ds sin t

Question: Let V = C ([a, b|) be the vector space consisting of all functions f(t) which are defined and continuous on the interval 0 ¤

Let V = C ([a, b|) be the vector space consisting of all functions f(t) which are defined and continuous on the interval 0 ‰¤ t ‰¤ 1. Which of the following conditions define subspaces of V? Explain your answer.
(a) f(0) = 0
(b) f(0) = 2/(l)
(c) f(0)f(l)= 1
(d) f(0) = 0 or f(l) = 0
(e) f(I - r) = tf(t)
(f) f(1-t)=l- f(t)
(g)

(h)

(i)

f(1)=I f(t)dt (t-1)f(r)dt = 0 f (s) sin s ds = sin t .

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