Question

# 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)

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

2021-01-11
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:
$$\displaystyle{T}{\left({u}\ +\ {w}\right)}={T}{\left({u}\right)}\ +\ {T}{\left({w}\right)}$$
$$\displaystyle{T}{\left({k}{u}\right)}={k}{T}{\left({u}\right)}$$
Calculation:
For T is a linear transformation, that is $$\displaystyle{T}:{C}{\left[{0},{1}\right]}\rightarrow{C}{\left[{0},{1}\right]}$$ defined by $$\displaystyle{T}{\left({f}\right)}={g}{]}{w}{h}{e}{r}{e}{\left[{g{{\left({x}\right)}}}={e}^{{{x}}}{f{{\left({x}\right)}}}\right.}$$
Such that assume g is also present in the domain of $$\displaystyle{T}:{\left[{0},{1}\right]}\rightarrow{C}{\left[{0},{1}\right]}$$, that means $$\displaystyle{T}{\left({g}\right)}={f}$$ where $$\displaystyle{f{{\left({x}\right)}}}={e}^{{{x}}}{g{{\left({x}\right)}}}$$
Then,
$$\displaystyle{T}{\left({f}\right)}={e}^{{{x}}}{g}$$
$$\displaystyle{T}{\left({g}\right)}={e}^{{{x}}}{f}$$
$$\displaystyle{T}{\left({f}\right)}\ +\ {T}{\left({g}\right)}={e}^{{{x}}}{g}\ +\ {e}^{{{x}}}{f}$$
$$\displaystyle={e}^{{{x}}}{\left({f}\ +\ {g}\right)}$$
$$\displaystyle={T}{\left({f}\ +\ {g}\right)}$$
Thus it shows that $$\displaystyle{T}{\left({f}\ +\ {g}\right)}={T}{\left({f}\right)}\ +\ {T}{\left({g}\right)}$$, that means it satisfies the addition property
Now, take any scalar be k
And,
$$\displaystyle{T}{\left({k}{f}\right)}={e}^{{{x}}}{\left({k}{f}\right)}$$
$$\displaystyle={k}{e}^{{{x}}}{f}$$
$$\displaystyle={k}{\left({e}^{{{x}}}{f}\right)}$$
$$\displaystyle={k}{T}{\left({f}\right)}$$
Thus is also satisfies the scalar property.
So, it satisfies all the condition of the linear transformation.
Hence, the expression of $$\displaystyle{T}:{C}{\left[{0},{1}\right]}\rightarrow{C}{\left[{0},{1}\right]}$$ defined by $$\displaystyle{T}{\left({f}\right)}={g}$$ where $$\displaystyle{g{{\left({x}\right)}}}={e}^{{{x}}}{f{{\left({x}\right)}}}$$ is linear transformation.
As for $$\displaystyle{T}:{C}{\left[{0},{1}\right]}\rightarrow{C}{\left[{0},{1}\right]}$$, the property of addition and scalar are satisfied. Thus the expression of $$\displaystyle{T}{\left({f}\right)}={g}$$ where $$\displaystyle{g{{\left({x}\right)}}}={e}^{{{x}}}{f{{\left({x}\right)}}}$$ is linear transformation.