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

linear algebramatricesrepresentation-theory

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

enter image description here

enter image description here

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?

Best Answer

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 $A=F'(0)$.

Related Question