I'm reading K. Muller's Introduction to Supersymmetry about spinor representation. He said that the components of a spinor are Grassmann variables.
I understood Grassmann variables as follows. For a vector space $V=\rm{span}\{\theta_1,\cdots,\theta_n\}$, the direct sum $$ \Lambda(V)=\mathbb{C}\oplus V\oplus(V\wedge V)\oplus\cdots\oplus\wedge^nV $$ forms an algebra under wedge product $\wedge$.
On the other hand, Muller constructed Weyl spinors as elements of two representation spaces $\psi\in F$ and $\bar{\psi}\in\dot{F}$. Their components $\psi_1,\psi_2,\bar{\psi}_{\dot{1}},\bar{\psi}_{\dot{2}}$ are Grassmann variables. It seems that he meant these $\psi_i\in V$ instead of $\psi_i\in\Lambda(V)$.
My question is that what's the dimension of $V$ here $n=?$ At first I thought $n=4$ and $V=\rm{span}\{\psi_1,\psi_2,\bar{\psi}_{\dot{1}},\bar{\psi}_{\dot{2}}\}$, but $$ (\sigma^0)_{A\dot{B}}\epsilon^{\dot{B}\dot{C}}\bar{\psi}_{\dot{C}}=\psi_A $$ seems to suggest that $\psi_A$ and $\bar{\psi}_{\dot{A}}$ are linear dependent. But if $n=2$, the expressions like $(\theta\psi)(\theta\phi)$ will simply be $0$?
