2023年6月8日 · A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else.

For me affine spaces are useful mainly because of their affine transformations, that is of bijective transformations, preserving straight lines of the affine space. Affine thansformations of $\Bbb R^n$ are less rigid than motions, because we don’t required to keep distances. So, for instance, we can affinely transform a circle into an ellipse ...

Recently, I am struglling with the difference between linear transformation and affine transformation. Are they the same ? I found an interesting question on the difference between the functions. ...

So, having the definition of an affine set, we can construct the appropriate parallel subspace. Of course ...

2017年6月24日 · To augment Lord Shark's answer, I just wanted to talk a little about the intuition behind it. Intuitively, a set of vectors is linearly dependent if there are more vectors than necessary to generate their span, i.e. the smallest subspace containing them.

$\begingroup$..on an affine space is the underlying vector space, which gives you the ability to add vectors to points and to perform affine combinations; this is something not available on a general Riemannian manifold. I do agree that you have a way to turn an affine space into a Riemannian manifold (by means of non canonical choices).

I am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me.

2017年5月2日 · Note that an affine hyperplane, differently than a hyperplane, needs not pass through the origin (and thus, somewhat confusingly, an affine hyperplane is not a hyperplane). Let us refer to the set of such points as the affine span, and denote it with $\operatorname{aff}(\{\mathbf p_i\}_{i=1}^m)$.

Note that the second definition is a generalisation of the first. A set is affine iff it contains all lines through any two points in the set (hence, as a trivial case, a set containing a single point is affine). (Thanks to @McFry who caught a little sloppiness in my original answer.)

It is my hunch that affine independence is analogous to linear independence in that, a set of vectors is (affinely/linearly) independent if none of the vectors is an (affine/linear) combination of the others. If this is the case, then what does the condition $\lambda_{1} +... +\lambda_{n} = 0$ have to do with anything?