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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2
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0answers
28 views

Dependence of linear algebra theorems of the commutativity of the field.

In the linear algebra course I took vector spaces where introduced with a (commutative) field. The classical theorems are proven under this assumption. However, I was wondering what implications it ...
0
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0answers
30 views
+50

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
0
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0answers
10 views

Two meanings to affine independence? (help me clear up my misunderstanding)

I must be misunderstanding something. Let's look at the following two definitions for a set of points $S=\{v_1,v_2,...,v_k\}$ to be affinely independent: 1) S is affinely independent if the set ...
6
votes
1answer
49 views

Intersection of affine varieties is affine

Let $M,N\subset\mathbb{P}^n$ quasiprojective varieties such that there exist isomorphisms $i\colon M\rightarrow Z(a)\subset \mathbb{A}^m$ and $j\colon N\rightarrow Z(b)\subset \mathbb{A}^m$ for ideals ...
0
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0answers
29 views

Points are affinely independent $\iff$ No Hyperplane that contains all points

I want to prove that the following statements are equivalent: (1) The points $P = \{p_1, ..., p_n\}$ are affinely independent (2) For every point $p_i \in P$ there exists a Hyperplane $H$, ...
0
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1answer
15 views

Prove that the ratio of lengths of parallel segments is invariant under affine trasnformations

The complete question is as follows: Prove that under an affine transformation the ratio of lengths on parallel line segments is an invariant, but that the ratio of two lengths that are not ...
0
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0answers
10 views

Does a simplex uniquely determine an affinely indepedent set which it is a convex hull of?

Let's say you're given an $n$-simplex, and you're told that it is the convex hull of two affinely independent sets $A_1$ and $A_2$. Do $A_1$ and $A_2$ need to be equal? To rephrase the question, given ...
5
votes
1answer
926 views

Affine Independence $\iff$ Linearly Independent

I guess I'm having some trouble getting my head around the notion of affine independence. As I've been taught, a set of vectors $\{\vec{x_1},\ldots,\vec{x_n}\}\subset \mathbb{R}^d$ is affinely ...
1
vote
1answer
30 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
0
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0answers
3 views

orientation induced by embedded simplex

let $\Delta_n$ be an affine simplex, we fix an orientation on it as the ordered set of vertices $\{A_0,\dots , A_n\}$. Now linearly embed it inside $\mathbb{R}^n$. According to Milnor-Stasheff ...
0
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1answer
32 views

Check if ray intersects internals of $D$-facet

Given a ray $\overrightarrow{r_0} + \overrightarrow{v} \cdot t, t \in [0;+\infty)$ and a $(D - 1)$-simplex, defined by $D$-tuple of its vertices $p_i = (p_i^1, p_i^2, \dots, p_i^D), i \in \{1, 2, ...
1
vote
1answer
32 views

Show that the variety $V(I(X))=X$

In the ring $R=K[x_1,...,x_n]$, the variety of an ideal is defined as $V(I)=\{(a_1,...,a_n)\in K^n|f(a_1,...,a_n)=0, \space\forall f\in I\}$ The ideal of a variety is defined as $I(V)=\{f\in ...
0
votes
1answer
17 views

Prove that set is in general position

Let $A$ be not empty set of points in general position contained in affine space $H \subset K^n$ . Let $q \in H-A$(where - is set complement) Prove set $A \cup \{q\}$ is in general position iff $q ...
1
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0answers
17 views

Octonions - affine space

I'm writing a project on Cayley's algebra. I have some topics which I have to follow and I've managed to solve most of them,except 2. I have written about their rule of multiplication,together with ...
2
votes
1answer
40 views

Dense basic open set contained in dense open subset

For an affine variety $X$ with coordinate ring $A$ it is not hard to see that for $g\in A$ the basic open set (or distinguished open set) $$D(g):=\{ P\in X | g(P)\neq 0\}$$ is dense in $X$ if and only ...
0
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0answers
16 views

transform orthonormal coordinate system to another

I have one orthonormal coordinate system ABC that it's origin is the point p0. I would like to transform it to another orthonormal coordinate system A'B'C', that it's origin is p1. I know how to ...
1
vote
0answers
9 views

Proving Pappus' theorem in a finite Affine Geometry

Let $\mathcal{A}$ be an affine plane with a finite amount of points on each line. Suppose that Desargues' theorem holds in $\mathcal{A}$. Then it is known that we can associate a division ring ...
5
votes
2answers
103 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
3
votes
1answer
65 views

Relationships between affine closures and convex closures

Let $V$ denote a vector space. Then the following concepts make sense: affine subset of $V$ affine closure (affine "hull") of a subset of $V$ Suppose $V$ is in fact a real vector space. Then the ...
11
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4answers
2k views

What are affine spaces for?

I'm studying affine spaces but I can't understand what they are for. Could you explain them to me? Why are they important, and when are they used? Thanks a lot.
4
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0answers
27 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
2
votes
1answer
45 views

Affine Transformations: Book to Study over the Summer

I've briefly heard of affine transformations in both linear algebra and calculus and I'd like to find a good book on the subject to study over the summer. So what's a good undergrad-level book on ...
2
votes
0answers
13 views

Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but ...
1
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0answers
10 views

Representation of Affine Maps

I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$: $$ f[\lambda x+ (1-\lambda) y]=\lambda ...
3
votes
1answer
201 views

Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, ...
0
votes
0answers
23 views

Finding all the invariant subspaces of a certain linear transformation.

Assuming I have given affine transformation $ \mathbb{R}^3\to \mathbb{R}^3 $ which has matrix representation $$ \left[\begin{array}{cccc} 3&2&-3&-10\\ 4&10&-12&-29\\ ...
1
vote
1answer
58 views

Vector bundles on $\mathbb{A}^1_k$ with doubled origin?

One of the most common examples of gluing affine lines is the affine line $\mathbb{A}^1_k$ with doubled origin. Out of curiousity, is there a known classfication of the vector bundles on this space?
3
votes
1answer
339 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
1
vote
2answers
42 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [closed]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
3
votes
0answers
81 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
0
votes
1answer
15 views

how to find affine formula

I'm struggling with finding formula of affine transformation where we have: $(1,3) \to (3,5)$ $(2,1) \to (0,6)$ $(4,0) \to (0,11)$ I know in affine transformation we have ...
1
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0answers
15 views

Non-affinely parametrized geodesics

Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent ...
0
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0answers
15 views

Is an affine transformation s graph always a plane (or hyperplane)?

Say I have an affine map, f: R2 --> R. Will the graph of f be a plane in R3? What about for the general case of F:Rn-->Rq? Will the graph of F be a hyperplane?
0
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1answer
19 views

Distance from affine vector space?

I've got an affine vector space $W$ defined by a collection of vectors $\{v_1, v_2, ... v_n\}$. Each vector in that space could be represented as a sum of the form $\sum_{i=1}^n w_i * v_i$, where ...
0
votes
0answers
16 views

Prove that if a transformation sends centroids to centroids then it is affine.

I have to prove that if a transformation sends centroids to centroids then it is affine. I sort of have the second part of the proof: assuming that I know that a transformation that preserves ...
0
votes
1answer
11 views

Calculate original coordinate after changing the transformation matrix

I am working with HTML5 canvas : I apply 2 transformations : Translate my canvas to (x,y) Rotate it with an angle a. Then I draw a circle at the position (x1,y1) I calculated my transformation ...
1
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0answers
21 views

Vanishing points from three collinear points

I would like to find the 2D vanishing point from a three collinear points as is shown in "Multiple View Geometry in Computer Vision" Example 2.19 (see here). What I did so far: 1 - I've extracted ...
3
votes
1answer
41 views

Product of affine varieties is the product of topological spaces

Let $k$ be an algebraically closed field, and $A, B$ affine $k$-algebras. We can define a functor $\mathfrak F$ from the category of affine $k$-algebras to that of affine algebraic varieties, by ...
4
votes
1answer
128 views

What is an affine space?

I am having trouble understanding what an affine space is. I am reading Metric Affine Geometry by Snapper and Troyer. On page 5, they say: "The upshot is that, even in the affine plane, one can ...
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0answers
20 views

Do affine spaces have coordinate transformations?

I asked a question on Physics SE and there seemed to be some confusion as to whether affine spaces could have coordinate transformations. Specifically, the particular space I was working with was ...
5
votes
4answers
132 views

Term for similarity transformation which is not a translation

What's the best (i.e. most concise) term to refer to an orientation-preserving similarity transformation which is not a translation? Here are some descriptions I could think of, but all of them feel ...
3
votes
2answers
119 views

Theory and problems book in euclidean, affine, and projective geometry

Could you recommend a rich, clear, and complete theory book on euclidean, affine and projective spaces (i.e., "geometry"); and an interesting exercise book full of non-trivial problems and exercises?
0
votes
2answers
60 views

Identify a quadric

Could you tell me how to identify a given quadric? Given a conic section, I should find an orthonormal affine frame in $\mathbb{R}^2$ (with standard dot product) in which the equation has a canonical ...
0
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0answers
18 views

Error metric from affine transformation

I have an affine transformation matrix consisting of a translation and a rotation of a 3D object. I'm developing an algorithm where such a translation should ideally converge to identity, i.e. any ...
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0answers
22 views

Find a normal for an affine hull

How do I find a normal for the affine hull of {[3,1,4], [5,2,6], [2,3,5]}? Thanks
1
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0answers
23 views

Determining if a set is projective or not

In $\mathbb{P}^3$ define the following sets: $$X=\{w_0w_1^2=w_2^2w_3-w_3^3\}\\Y_1=\{y_3=0\}\\Y_2=\{\sum_{i=0}^3 w_i=0\}\\Y_3=\{w_0+w_1+w_2+2w_3=0\}$$ Does the set $Z=X\cap Y_3\setminus((X\cap ...
3
votes
0answers
35 views

Calculate the singular points of affine curve

I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$ The point $P=(x,y)$ is singular $\Leftrightarrow$ If $x=0$ we find $y=0$ and then from the ...
0
votes
0answers
38 views

$Ax = b$ & $Ax + b$

Ask a dumb question but confuse me long time. The following is what I know: 1st case $Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$. 2nd case $ f(x) = Ax + b$ ...
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0answers
12 views

Explain how lines and points in the 2D plane form an affine plane?

I think I understand the affine transformation, but I just have trouble describing how lines and points in the 2D plane form an affine plane.
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0answers
20 views

help me to check the following argument on a linear transformation

all, Just want to confirm the following argument: Assumption: we know that $S:=\{(x_1,x_2,x_3) : \|(x_1,x_2)\|_2 \leq (1+k)x_3\}$, for a given value $k$. By linear transformation: ...