Consider the dynamical system $\dot{x}=F(x,\lambda), x\in\mathbb{R^n}$, and let $F(0,\lambda)=0$ for some neighborhood of $\lambda_{0}$, the transcritical bifurcation arises if we have $w\frac{\partial F}{\partial x\partial\mu}(v)\neq 0$ ,where $w$ and $v$ are the left and right eigenvector of $F$ for eigenvalue $0$. I want to know what it means geometrically.
On the other hand, in the setting of Banach space, the condition changes to $\frac{\partial F}{\partial x\partial\mu}(v)\notin R(D_xF(0,\lambda_0))$, since we can also interpret $w$ above as $Ker(F^*)$, and $Ran(F)\perp Ker(F^*)$, so I believe that's the Banach space case says, I want to know if it is right
