Solving Linear Algebra Homework: T(e^5x), T(3e^4x), etc.

[andrassy]

Homework Statement

Let F be the vector space of all functions mapping R into R and let T: F -> 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 the data.

a. T(e^5x)
b. T(3e^4x)
c. T(3 + 5e^3x)
c. T((e^4x + 2e^5x)/e^2x)

Homework Equations

None

The Attempt at a Solution

I know that a linear transformation preserves vector addition and scalar multiplication. To get e^5x, I need to multiply e^2x and e^3x. Would this be scalar multiplication? I figured it would not be because they are vectors in this circumstance. The other problems are similar. Can I multiply them? if so, why?

 

[HallsofIvy]

I see no attempt on your part to solve these problems. Do you want us to do them for you so you can trick your teacher into not actually teachin g you how to do them and then failing the test?

 

[andrassy]

I see no attempt on your part to solve these problems. Do you want us to do them for you so you can trick your teacher into not actually teachin g you how to do them and then failing the test?

Wow well clearly I don't understand the concept involved. I tried to say what I knew/understood. I don't want anyone to do the problems for me I just want a hint or some help how to start so I can at least attempt the problem

 

[jjou]

Using scalar multiplication and vector addition alone, you should at least be able to work out (c) and (d).

 

[eys_physics]

The exercise is not to calculate them all.
"Find the following, if it is determined by the data." i.e.
So when you have to answer when are they determined by data?
The answer is that they are determined when they can be written as a linear combination of
$$1, exp(2x), exp(3x)$$
If that is the case you can use the linerarity of T.