I've just started reading what seems to be a fairly elementary introduction book on Lie algebras. I don't know anything about the subject- I picked it up mainly because the preface suggests linear algebra would be the only perquisite- So I'm very sorry if the question is too elementary this is the first time I'm reading mathematics book on my own and trying the exercises just for fun.
I need to show 2 properties from first principles:
- Show that $[v,0]=0=[0,v],\,\forall v\in V$
- Suppose that $x,y\in V$ satisfy $[x,y]\neq 0$. Show that $x$ and $y$ are linearly independent
Here's how I proved them:
- Note that $0=v-v$ so $[v,v-v]=[v,v]-[v,v]$ by the bilinearity, and from the property of the lie brackets $[v,v]=0$ we get $[v,0]=0$, the other side should be much the same because of the bilinearity.
- I will assume $x,y$ are linearly dependent and derive a contradiction. Since $x,y$ are linearly dependent there exist $\alpha \in \mathbb{F}$ such that $y=\alpha x$ so we can write $[x,y]=[x, \alpha x]=\alpha[x,x]=0$, the reasoning are very similar to 1.
Are these proofs fine? Are there any other proofs for these properties that I'm missing?
Thanks!
Both proofs are correct. However, the first statement holds for any bilinar map, not just Lie brackets. In fact, if $B$ is such a bilinear map,$$B(v,0)=B(v,0+0)=B(v,0)+B(v,0)$$and therefore $B(v,0)=0$.