Whether T is a linear transformation, that is T : C[0,1] rightarrow C[0,1] defined by T(f)=g, where g(x)=e^{x}f(x)

Bevan Mcdonald 2021-01-10 Answered
Whether T is a linear transformation, that is T:C[0,1]C[0,1] defined by T(f)=g, where g(x)=exf(x)
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Expert Answer

faldduE
Answered 2021-01-11 Author has 109 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:C[0,1]C[0,1] defined by T(f)=g]where[g(x)=exf(x)
Such that assume g is also present in the domain of T:[0,1]C[0,1], that means T(g)=f where f(x)=exg(x)
Then,
T(f)=exg
T(g)=exf
Perform additional operation.
T(f) + T(g)=exg + exf
=ex(f + g)
=T(f + 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)=ex(kf)
=kexf
=k(exf)
=kT(f)
Thus is also satisfies the scalar property.
So, it satisfies all the condition of the linear transformation.
Hence, the expression of T:C[0,1]C[0,1] defined by T(f)=g where g(x)=exf(x) is linear transformation.
Answer:
As for T:C[0,1]C[0,1], the property of addition and scalar are satisfied. Thus the expression of T(f)=g where g(x)=exf(x) is linear transformation.

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