Showing that a matrix F is a matrix representation of $\mathbb{R}$

Question

The question is given below and its answer are given below:

1-I know that according to Vinberg book which is called "Linear representations of groups" all finite dimensional differentiable representations of thereals under addition are given by sending t to exp(tA) where A is a linear operator on the vector space of given dimension " but how can this information help me here?

2- My professor mentioned that the matrix F in part(a) contain a typo where it is $\sinh t$ instead of $-\sinh t$.

3-The definition of A MATRIX REPRESENTATION of the group $G$ over the field $K$ is a homomorphism of $G$ into the group $GL_{n}(K)$ of invertible matrices of order n over $K$. So what are the steps that I should do to show that $F$ is the matrix representation of $\mathbb{R}$ and also what are the steps I should do to find the matrix $A$?

My suggestions:

1- For the first part I should show that $F(t) = e ^ {t A}$ is linear but how? and I should show that F(only not F(t)) is a homomorphism but also how ?

2- For the second part I do not know what to do at all.

Could anyone help me in removing this discrepancies?

Answer 1

One of the way to show that one group represents another is to find a bijection from one group to another and show that the bijection keeps the group operation. Since you are already given a function $F(t)$, you want to show that

1. it's reversible: $F(t_1)=F(t_2)$ implies $t_1=t_2$
2. it keeps the group operation: $F(t_1)F(t_2) = t_1+t_2$

Part 2. After you have shown the above, you know that $F(t)=e^{tA}$. So one of the tricks how to find $A$ is to take Taylor Series around $t=0$:

$$ F(t) = F(0) + tF'(0) + \frac{t^2}2F''(0)+\ldots,\qquad e^{tA}=I + tA +\frac{t^2}2A^2+\ldots, $$

from which we know that $A=F'(0)$.