If R be the additive group of real numbers and R+ the multiplicative group of positive real numbers show that the following mapping are isomorphism_ (i) f: R→ R+ s. t. f(x) = ex, x € R. (ii) f: R+ → Rs. t. f(x) = log x, XER+.

Let T : R² → P2(R) and U : Rª –→ M2x2 (R) be lincar transformations. A student claims T cannot be invertible because dim(R²) + dim(P2(R)). If the student is correct, prove their claim. If the student is not correct, explain why and give an example to illustrate. Clearly state whether or not the student is correct as part of your solution.

Prob. 1. Consider the group of functions (R) under addition of functions (so (f+g)(x) = f(x)+g(x)). Determine the elements of the cyclic group (f), where f(x) = x-1.