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**1. 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)

**2. Homework Equations**None

**3. 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?