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

postillan4 2021-01-28 Answered
Whether T is a linear transformation, that is T:C1[1,1]R1 defined by T(f)=f(0)
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Expert Answer

falhiblesw
Answered 2021-01-29 Author has 97 answers
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 gC1[1,1]R1
Then,
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 k
And
T(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.
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