Even knowing the definition of affine subspace, how can I work the definition to prove rigorously what is or isn't an affine subspace? Right now I can see what is or isn't, but I couldn't explain it on an exam, for example.
Examples:
$\{(a,b,c) \in \mathbb{R}^3 : b = c,\space a\ge 0\}$
This isn't an affine subspace. My ideia is that it isn't even a vectorial space even though it goes through $(0,0,0)$, but I can't formalize.
$\{(a,b,c) \in \mathbb{R}^3 : a+b = 2\}$
This is an affine subspace. It's a plane. How do I formalize? Also, how do I show the dimension is 2?
Other example is a sphere in $\mathbb{R}^3$, of course it is not an affine subspace, but what should I write down to say that it isn't?
Thank you!