# Tagged Questions

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.

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### Criterion for an affine isomorphism.

I am reading Don Taylor's book 'The Geometry of Classical Groups' and currently I am trying to understand the affine geometry section. There is a lemma which appears to be a criterion for a bijection ...
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### (Reference Request) Proofs for basic facts about regular functions on algebraic sets.

I am writing an assignment about algebraic and analytic sets in $\mathbb{C}^n$ and, when searching for references, came across the book Algebraic Geometry III. The book is a bit out of my depth, yet ...
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### Scaling in world space

In a hierarchical transformation system, where a node has one parent and children (Tree form) I want to scale an object with respect to world space axis. My transformation order is (Translate * Rotate ...
1answer
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### matrix transformation of deformed rectangle

I am working on touch screen calibration, and have come across a problem. My area of touch screen input is a Trapezoid which looks like a square on one side and a triangle on the other. (the angle ...
2answers
105 views

### Proof verification affine curve not isomorphic to plane curve

I'm trying to prove that the affine curve $X\subset\mathbb{A}^3$ given by $\alpha:\mathbb{A}^1\to\mathbb{A}^3$, $t\mapsto(t^3,t^4,t^5)$, is not isomorphic to a plane curve. Here is what I've done: it ...
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### Most general space on which we can do calculus

I have two somewhat related questions: Question 1: What is the most general space (set of objects) on which we can do calculus? Is it a normed space, or can we relax the conditions a bit further? ...
1answer
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### Isomorphisms and orthogonality in projective space.

I've been studying projective space in algebraic geometry for a few days from Perrin's 'Algebraic Geometry: An introduction'. In the first page of chapter III it reads ...projective space $P^n$ ...
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### How many convexly independent vectors there are in $\mathbb{R}^n$

I know there are n linearly independent and n + 1 affinely independent vectors in $\mathbb{R}^n$. But how many convexly independent there are? I think there are infinity number of them because if I ...
2answers
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### Give an example of an affine space that is not a vector space

We know that any vector space is an affine space, but can you give an example of an affine space which is not a vector space? I don't know any such examples. This is an interview question, not ...
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### Book(s) about Affine geometry.

A quick look on Stack Exchange enabled me to discover "Geometry" from Michele Audin which is very close from what I'm expecting but there isn't the correction of the exercices. To be more specific, I'...
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1answer
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### Dimension of product of affine varieties

Let $X\subset \mathbb{A}^n_K$ and $Y\subset \mathbb{A}^m_K$ be affine varieties. How can I prove that dimension of the product variety $X\times Y \subset \mathbb{A}^{m+n}_K$ is dim$X$+dim$Y$? Here I ...
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### Show that in finite affine geometry all lines are on the same number of points step by step proof.

i'm wondering if someone could please give me a step by step proof of the following problem Show that in finite affine geometry all lines are on the same number of points
1answer
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### Convert affine coordinates to projective coordinates?

For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in ...
1answer
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### Every line in $\mathbb{R}^2$ can be described…

I came across such a statement: Let $A = \mathbb{R}^2$, $a,b,c \in A$ be points (we treat $\mathbb{R}^2$ as an affine space). Then any line $L \in A$ can be described as L = \lbrace s_1 a + ...
1answer
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### affine variety/space vs. toric variety

I think I'm not quite clear on the meaning of a toric variety... Could someone explain the relation/difference between the affine variety/space and the toric variety? I know that affine ...
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### Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2.

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2. Here's my attempt at a solution: We know that any rotation in $\mathbb{R^3}$ can be ...
2answers
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### The composition of two homotheties is either a homothety or translation

Let $f$ and $g$ be two homotheties that don't have the same center. What kind of affine transformation is their transformation? I know it can be either a homothety or translation, but I don't know ...
1answer
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### Affine variety as subset of another affine variety

I am trying to understand why the following statement is true: If $S$ and $S'$ are subsets of $\mathbb{K}[X_1,...,X_n]$ such that $S\subseteq S'$, then $\mathcal{V}(S') \subseteq \mathcal{V}(S)$....
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### Existence of affine hull of set S

"Obviously, the intersection of an arbitrary collection of affine sets is again affine. Therefore, given any $S \subset R^{n}$ there exists a unique smallest affine set containing $S$ (namely, the ...
1answer
49 views