# geometrical interpretation of $\mathbb{Z}/2\mathbb{Z}$ graded space

According to wikipedia, a $\mathbb{Z}/2\mathbb{Z}$ graded space (super vector space) $V$ is a a vector space which can be decomposed in a direct sum $V=V_0 \oplus V_1$ where elements of $V_0$ are called even and elements of $V_1$ are called odd. First of all I don't understand how this definition is relied to Grassmann coordinates? Secondly do we have a geometrical interpretation of $V_1$, I mean can I draw two vectors of $V_1$ on a sheet?

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Who said this definition should be related to Grassmann coordinates? This is just a very bland general definition. For any vectorspace whatsoever you can choose a direct sum decomposition into $V_0$ and $V_1$, and call the resulting graded vector space a super vector space. If the vector space is over $\mathbb R$ and the dimension of the subspace $V_1$ is at most $2$, you can draw $V_1$ on a sheet (in other cases this is more problematic), but that is totally unrelated to the notion of a graded vector space. –  Marc van Leeuwen May 6 '12 at 11:18
Super vector space in physics are directly related to grassmann coordinates but I try to understand more than just formal physicist's definition. In fact I think that super vector space are related to grassmann coordinates if the $K$ fiels is an grassmann algebra, but maybe it is wrong? –  PanAkry May 6 '12 at 11:50