So I am havig an exam, and one of the examples last year was this:
IN vector space $R^3$, find intersection of two affine subspaces:
$$ (2,1,1,3,-3) + Span{(2,3,1,1,-1)}$$
and
$$N^T\cdot(x-p)=o\quad where \quad N=\begin{pmatrix}-2 & -1 & 0 & -1\\\ 1 & 0 & 0 & 0\\\ 0 & 1 & 0 & 0\\\ 0 & 0 & 1 & 0\\\ 0 & 0 & 0 & 1\end{pmatrix} \quad and\quad p=(1,1,2,1,2)$$
Now from what I know about affine spaces, I guess that in the first subspace (the one with span in it) the first round bracket is a point and the round bracket in span, is a translation vector.
However problem is with a second subspace, given by matrix equation, I can easily calculate x, sice it is in fact p, but that is just one round bracket, so is it suppose to be a point without vector? And if yes, then there is another problem, because when I create this equation to find intersection: $$(2,1,1,3,-1)\ +\alpha(2,3,1,1,-1)\ = \ (1,1,2,1,2)$$ it is overdetermined so it has no solution.
Where am I making a mistake?