Find the subspace of $\mathbb{C}^n$ with given Plücker coordinates. I'm studying Representation theory and I'm trying to understand the Grassmanian and Plücker coordinates. This problem is an exercise from Young Tableaux: With Applications to Representation Theory and Geometry

Find the subspace of $\mathbb{C}^4$ with Plücker coordinates $x_{1,2}=1, x_{1,3}=2, x_{1,4}=1, x_{2,3}=1, x_{2,4}=2$ and $x_{3,4}=3$

I've seen a few different examples but the book only gives the vague idea and I'm having trouble understanding how I should go about finding the subspace they are asking for.
 A: To start with $x_{2,3} x_{1,4}-x_{1,3}x_{2,4}+x_{1,2}x_{3,4}=1\cdot 1-2\cdot 2+1\cdot 3=0,$ the Plücker relation is satisfied. This relation gives the Plücker embedding of the grassmannian $G(2,{\Bbb C}^4)$ of $2$-subspaces in ${\Bbb C}^4$ as a quadric in ${\Bbb P}^5.$
Now the Plücker coordinates $(x_{1,2}:x_{1,3}:x_{1,4}:x_{2,3}: x_{2,4}:x_{3,4})$ are the $2\times 2$-minors of $\begin{pmatrix} a_1&a_2&a_3&a_4\\b_1&b_2&b_3&b_4\end{pmatrix}$ i.e.
$x_{1,2}=\begin{vmatrix} a_1&a_2\\b_1&b_2\end{vmatrix}=1,$
$x_{1,3}=\begin{vmatrix} a_1&a_3\\b_1&b_3\end{vmatrix}=2,$
$x_{1,4}=\begin{vmatrix} a_1&a_4\\b_1&b_4\end{vmatrix}=1,$
$x_{2,3}=\begin{vmatrix} a_2&a_3\\b_2&b_3\end{vmatrix}=1,$
$x_{2,4}=\begin{vmatrix} a_2&a_4\\b_2&b_4\end{vmatrix}=2,$
$x_{3,4}=\begin{vmatrix} a_3&a_4\\b_3&b_4\end{vmatrix}=3.$
Taking the ansatz $a_4=b_4=1$ and solving $a_1b_2-b_1a_2=1,a_1b_3-b_1a_3=2,a_1-b_1=1,a_2b_3-b_2a_3=1,a_2-b_2=2,a_3-b_3=3$
you get e.g. $\begin{pmatrix} -1/2&0&1/2&1\\-3/2&-2&-5/2&1\end{pmatrix}$ and the subspace you want is the span of the row vectors.
