Linear Algebra Problem: Solving for T in F(R) with Given Functions

[Nothing000]

Let F be the vector space of all functions mapping $$R$$ into $$R$$, and let $$T:F\rightarrow F$$ be a linear transformation such that $$T(e^{2x})=x^{2}$$, $$T(e^{3x})=sinx$$, and $$T(1)=cos5x$$. Find the following, if it is determined by this data.

$$T(e^{5x})$$
$$T(3e^{4x})$$
$$T(3+5e^{3x})$$
$$T(\frac{e^{4x}+2e^{5x}}{e^{2x}})$$

 

[Nothing000]

I have no idea what to do here. Does this involve the kernel of T?

 

[radou]

I have no idea what to do here. Does this involve the kernel of T?

Try using the properties of a linear transformation: http://mathworld.wolfram.com/LinearTransformation.html" .

 

[Nothing000]

So am I supposed to factor out the $$e^{x}$$ term like this

$$T(e^{3x})=sinx$$
$$e^{x}T(e^{2x})=sinx$$
$$T(e^{2x})=\frac{sinx}{e^{x}}$$

and since $$T(e^{2x})=x^{2}$$

it must be true that $$\frac{sinx}{e^{x}}=x^{2}$$

Am I on the right track?

 

[Nothing000]

Oh wait, that can't be right, because $$e^{x}$$
is not a constant. Duh.

 

[Nothing000]

So the only one that I actually can figure out is
$$T(3+5e^{3x})$$
since it is preserved by scaler multiplication and vector addition. Right?

$$T(3+5e^{3x})=T(3)+T(5e^{3x})$$

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =3T(1)+5T(e^{3x})$$

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =3cos5x+5sinx$$

Is that right?

 

[radou]

Yes, looks okay. Hint regarding the last one: $$T(\frac{e^{4x}+2e^{5x}}{e^{2x}}) = T(\frac{e^{4x}}{e^{2x}}+\frac{2e^{5x}}{e^{2x}})$$.

 

[Nothing000]

$$T(\frac{e^{4x}+2e^{5x}}{e^{2x}}) = T(\frac{e^{4x}}{e^{2x}}+\frac{2e^{5x}}{e^{2x}})$$
Is this what you mean?

 

[Nothing000]

Thank you so much bro. I REALLY apreciate the assistance. I owe you one. Gotto go to my L.A. class now. THanks.