Linear Algebra:Vector Subspace-2(B.Sc. Hons. Part-III ) By Dr P R PARIHAR

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Linear Algebra:Vector space, Subspace.

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Ans.1(a)- (i), (iii) option are correct they belongs to the set W, which is the subset of V.
(b).W is not subspace of V.
Because ax+by not belongs to W, where a, b belongs to R and x, y belongs to W.
Ans 2 W is not a subspace of V.
For eg. Let f and g are two continuous function and belongs to W. Let f(0)=3, g(0)=2 and a, b belongs to C[-1, 1] let a=1/2 and b=2/3 so, (af+bg)(0)=17/6 not an integer so af+bg not belongs to W
So W is not subspace of V

RishabhSharma-llwo

Ans 1 (a) . Option (i) and (iii) belongs to W.
(b) . W is not a subspace of V(R) since aX + bY doesn't satisfy the given condition
Ans 2.) W is not a subspace of C[-1, 1].
Let f(x) = 4x + 1 (belongs to W since f(0) = 1 is an integer)
Let g(x) = x + 2 ( belongs to W since f(0) = 2 is an integer)
Let a = 3/2 (belongs to R)
Let b = 1 (belongs to R)
Now p(x) = af(x) + bf(x)
Here p(0) = 7/2 which is not an integer
Therefore p(x) doesn't belong to W .

purvigarg

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