Determine whether or not the given set is a subspace of C(-infinity, infinity) (

Question

Determine whether or not the given set is a subspace of C(-infinity, infinity) (space of real valued continuous functions). S{all functions of the form a sin x + b cos x, where a and b are real number}.

Explanation / Answer

since asinx + bcosx will always be a real number => S is a subset of C ------ (1) for a=b=0, 0 belongs to S ----- (2) Let v1 belongs to S V1 = a1*sinx + b1*cosx Ler v2 belongs to S v2 = a2*sinx + b2*cosx let v3 = p*v1 + q*v2 (p and q are some real constants) v3 = p*(a1*sinx + b1*cosx) + q*(a2*sinx + b2*cosx) v3 = (p*a1 + q*a2)sinx + (p*b1 + q*b2)cosx say , p*a1 + q*a2 = A and p*b1 + q*b2 = B where A and B are new real constants v3 = A*sinx + B*cosx => v3 belongs to S Therefore, if v1,v2 belongs to S => p*v1 + q*v2 belongs to S -------(3) From (1), (2) and (3), S is a subspace of C

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