# Questions tagged [affine-geometry]

for questions about algebraic geometry that focus on affine space. For affine mappings in linear algebra (i.e. linear mappings plus translations), please use the linear-algebra tag or another appropriate tag.

65 questions
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### $\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
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### What is the difference between linear and affine function

I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated
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### Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
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### Is the complement of a complex affine algebraic set in an irreducible complex affine algebraic set (path) connected in the euclidean topology?

Let $n \ge 2$, and $V$ be an affine algebraic set in $\mathbb C^n$ and $W$ be an irreducible affine algebraic set in $\mathbb C^n$, with $V \subsetneq W$ ; then is it true that $W \setminus V$ is ...
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### Sphere to Ellipsoid affine transformation matrix

I am trying to find the minimum bounding box of an ellipsoid. In my search, I found this answer and also some other nice descriptions like this one to the problem. I am not a mathematician (I need ...
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### Affine space $A^n$ and definition of difference.

I'm not sure if this question would be more appropriate in Physics.SE, if so let me know. I need help in understanding this quote from "Arnold - Mathematical Methods in Classical Mechanics" (This is ...
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### What is the relationship between an affine space and an affine set?

I just learned what an affine space is and its basic properties. It is formally defined as $\Bbb A=(A,V,f)$ where $A$ is a set of points, $V$ is a vector space, and $f:A^2→V$ is a function st. for any ...
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### Basis of an affine subspace

Consider an affine subspace $D$ of an affine space or affine plane $\mathcal{A}$. Every set of points that are not elements of a proper affine subspace of $D$ is called a generating set of $D$. If ...
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### Is the group generated by $f(x)= x+2^{\nu_2(2x)}$ and $g(x)=2x$ a 2-dimensional affine space on $\Bbb Z[\frac12]\setminus 0$?

Does the free abelian group generated by composition of $f(x)=x+ 2^{\nu_2\left(2x\right)}$ and $g(x)=2x$ define a 2-dimensional affine space on $\Bbb Z\left[\frac12\right]\setminus0$? $2^{\nu_2(x)}$ ...
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### Affine plane of order 4?

I cannot seem to construct an affine plane of order 4. I have the construction for order 3- but cannot seem to come up with or find the construction for 4 anywhere. Could someone show me a picture of ...
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### dimension of quotient space

I am confused about the following: In Wiki: => dim(vector space) - dim(subspace) = dim(quotient space) In S. Boyd's textbook of cvx (p.22) => dim(subspace) = dim(affine set) Problem: As far ...
If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form $f(x)=ax+b$...