Question
Asked Sep 23, 2019
are not vector spaces?
23) Which of the following subsets of C[-1,1]
1
(x)dx 0
f(x) E CI-1,1]:
a)
-1
{f(x)e C-1,1]: f(-1) = f(l)}
{f (x) E C[-1,1]: f(x) 0 for xe[-1/2,1 / 2]}
{f(x) e Cl-1,1]: f (1) = 1}
e) {f(x)e CI-1,1]: f(l) = 0}
b)
c)
d)
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are not vector spaces? 23) Which of the following subsets of C[-1,1] 1 (x)dx 0 f(x) E CI-1,1]: a) -1 {f(x)e C-1,1]: f(-1) = f(l)} {f (x) E C[-1,1]: f(x) 0 for xe[-1/2,1 / 2]} {f(x) e Cl-1,1]: f (1) = 1} e) {f(x)e CI-1,1]: f(l) = 0} b) c) d)

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Expert Answer

Step 1

Condition for vector space:

It is known that, if a,be R and f.geS then af + bg e S
Then S is vector space.
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It is known that, if a,be R and f.geS then af + bg e S Then S is vector space.

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Step 2

a)

Consider the subset, 

1
=f(x)e C[-1,1f(x)dh = 0 }
1
1
Let f,gE S then ff(x)d = 0 and fg(x)dc = 0
-1
1
(af+ bg d=faf (x)dx bf(xyd«
-1
-1
a(0)b(0)
=0
Therefore, af bg ES.
Thus, the subset is vector space.
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1 =f(x)e C[-1,1f(x)dh = 0 } 1 1 Let f,gE S then ff(x)d = 0 and fg(x)dc = 0 -1 1 (af+ bg d=faf (x)dx bf(xyd« -1 -1 a(0)b(0) =0 Therefore, af bg ES. Thus, the subset is vector space.

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Step 3

b)

Consider the s...

S={f(x)EC[-1,1:f1)=S(1)} I
Let f,geS then f(-1) = f(1) and g(-1) g(1)
(afbg)-)af-1)+ ag(-1)
-af (1)+bg(1)
(af bg)()
Therefore, afbgeS
Thus, the subset is vector space.
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S={f(x)EC[-1,1:f1)=S(1)} I Let f,geS then f(-1) = f(1) and g(-1) g(1) (afbg)-)af-1)+ ag(-1) -af (1)+bg(1) (af bg)() Therefore, afbgeS Thus, the subset is vector space.

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