What does a manifold which is a solution to a set of $n$ affine equations look like?

Question

I was reading a paper by Duffie and Kan where they mention that the solution to $a_{ii}+b_{ii}.x=0, i=1 \text{ to } n$ is an $(n-1)$ dimensional manifold. I have never studied manifolds and if you could point me to a source which could help me get just the basic intuitive understanding it would be very helpful(In particular this sort of manifold)

Answer 1

My impression is it's just like a vector space, except that it doesn't have to contain the zero vector. So any translation of a vector space. That is, take a vector space, $V$, and consider all sums $v+w$, for $v\in V$ and some vector $w\not\in V$.

For manifolds there are many sources. I like Spivak, for instance.