The Plucker image is a closed projective algebraic set

For $r=0$ or $r=n$, the Grassmannian and $\mathbf{P}({\mathrm{\Lambda }}^{r}V)$ are both one point, so the assertion is immediate. Suppose henceforth that $0<r<n$. For ordered tuples $(i_1,\dots ,i_{r-1})$ and $(j_1,\dots ,j_{r+1})$, alternating multilinearity gives the signed Plucker relation ${\sum }_{a=1}^{r+1}(-1{)}^a{p}_{i_1\dots i_{r-1}{j}_a}{p}_{j_1\dots \widehat{j_a}\dots j_{r+1}}=0.$ The ordered-coordinate convention supplies every insertion sign and makes repeated indices contribute zero.

On the chart $p_{1\cdots r}\mathrm{\ne }0$, divide by that coordinate and use the relations with $I\subseteq \{1,\dots ,r\}$ to express every $p_I$ as the corresponding minor of the matrix $(I_r\mid A)$. Its row span has precisely those coordinates.

The analogous charts cover every nonzero coordinate point satisfying the relations. Thus every such point is decomposable, while step 1.1 proved the reverse inclusion; homogeneous quadratic equations make this locus closed, and the injective Plucker map realizes it as the required projective algebraic set.