Whether T is a linear transformation, that is T:C1[−1,1]→R1 defined by T(f)=f′(0)

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falhiblesw · Accepted Answer

Approach:For U and V vector spaces and T is a function from U to V, then T would be considered as linear transformation if for all u, w lies in U and all scalars k as if satisfies the following properties:T(u + w)=T(u) + T(w)T(ku)=kT(u)Calculation:For T is a linear transformation, that is T:C1[−1,1]→R1 defined by T(f)=f′(0)Such that assume g is also present in the domain of T:C1[−1,1]→R1, that means g∈C1[−1,1]→R1Then,T(f)=f′(0)T(g)=g′(0)Perform additional operation.T(f) + T(g)=f′(0) + g′(0)And,T(f + g)=(f + g)′(0)=f′(0) + g′(0)=T(f) + T(g)Thus it shows that T(f + g)=T(f) + T(g), that means it satisfies the addition property.Now, take any scalar be kAndT(kf)=(kf)′(0)=kf′(0)=k(f′(0))=kT(f)Thus is also satisfies the scalar property.So, it satisfies all the condition of the linear transformation.Hence, the expression of  T:C1[−1,1]→R1 defined by T(f)=f′(0) is linear transformation.Answer:As for T:C1[−1,1]→R1, the property of addition and scalar are satisfied. Thus the expression of T(f)=f′(0) is linear transformation.

Whether T is a linear transformation, that is T : C^1[-1,1] rightarrow R^1 defined by T(f)=f'(0)

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