When learning about linear subspaces in class we used this definition:
$$ \forall x,y \in K , \lambda , \mu \in \mathbb{R}: \lambda x+ \mu y \in K$$
For affine subspaces we used the following:
$$ \forall x,y \in K , \lambda , \mu \in \mathbb{R}, \lambda + \mu =1: \lambda x+ \mu y \in K $$
When I first saw these two definitions, I considered affine subsets to be more restricted than linear subsets because there was one more constraint ($ \lambda + \mu =1 $). So I was quite shocked when our professor told us that in $\mathbb{R^2}$ we can consider every line going through origin a linear subset, whereas an affine subset can be any line. So it seemed to me every linear subspace was affine, but not every affine subspace is linear. This was contradictory to my initial intuition from the definition because affine subspaces seemed to have one more constraint.
After thinking about it for a while a changed my perspective on the subject. I started to think you have more options with affine subspaces because the equation ($\lambda x+ \mu y \in K$) only needs to hold when $ \lambda + \mu =1 $. So in a way this constraint actually frees us to have more options. Now the equation doesn't have to be true for every $\lambda$ and $\mu$, it just has to be true when they add up to 1.
I thought I understood the topic very well. Until the definition for convex subspaces came along:
$$ \forall x,y \in K , \lambda , \mu \in \mathbb{R}, \lambda + \mu =1, \lambda, \mu > 0: \lambda x+ \mu y \in K $$
When I saw this I thought: "I know it! Being convex is being more free than affine! So every affine subset must be convex but convex subsets have more freedom (namely they only need to satisfy the equation when both $\lambda, \mu > 0$) so there should be some convex sets that are not affine!
But when the professor was giving examples for convex sets in $\mathbb{R^2}$ he said line segments. Now I am confused. Every affine subspace (lines in $\mathbb{R^2}$) have the line segments on them. So I cannot grasp how convex sets have more "freedom". Was my corrected intuition also wrong? Are all affine subspaces not convex? If my intuition is correct, can you give an example in $\mathbb{R^2}$ that is convex but not affine?
Thanks!
The issue here is that in the first comparison, you have compared a $2$-dimensional vector space (say, generated by two (independent) vectors $x,y$ that you started with) and a $1$-dimensional affine space (say, generated by two vectors $x,y$ that you started with). So constraints are constraints and they will shrink your space, as in the case of affine space and convex set. (In the latter comparison, starting with two independent vectors, both affine space and convex set have dimension 1).