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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Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
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14 views

Let $T(x)=Ax+v$, where $A\in GL (n,\mathbb{R})$ and $v \in \mathbb{R}^n$. Show that T sends affine subspaces of $\mathbb{A}^n$ to affine subspaces.

I'm not too familiar with Affine geometry so I feel as though I'm missing something from my solution: Let $E$ be an affine subspace defined by $E=\{P+ \textbf{v}\, |\, \textbf{v} \in U\}$ where $U$ ...
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1answer
33 views

Domain of definition of $(1-y)/x$ on $x^2+y^2=1$?

I'm self-teaching myself some basic algebraic geometry, and I wanted to double check something that seems too easy. An exercise sheet I found asks to compute the domain of definition of the rational ...
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7 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
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16 views

How to show that $F(m_0+v)=m_0+A(v), v∈V$ defines an affine map of $(M,V)$? [on hold]

Let$(M,V)$ be an affine space, and let $m_0 ∈ M, A ∈ L(V)$. I need to show that the equation $$F(m_0+v)=m_0+A(v), v∈V$$ defines an affine map of $(M,V)$ with linear part $LF = A$ that has $m_0$ as a ...
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19 views

Defination of Affine Space [on hold]

A. Let (S,V) be an affine space, and let $s_0 \in S$, $A \in L(V)$. Prove the following equation $$F(s_0+v) = s_0 + A(v), v \in V$$ is an affine map of (S,V) with linear part LF = A that has $s_0$ as ...
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1answer
22 views

Are the quasi-affine subsets of $\mathbb{A}^1_F$ necessarily open or closed?

For what follows, my definition of a quasi-affine subset of $\mathbb{A}^n_F$ is one which can be written as $Z_1\setminus Z_2$, where $Z_1$ and $Z_2$ are closed subsets of $\mathbb{A}^n_F$. (I think ...
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4answers
55 views

A function is convex and concave, show that it has the form $f(x)=ax+b$

A function is convex and concave, it is called affine function. That is the function: $$f(tx+(1-t)y)=tf(x)+(1-t)f(y),\, \, t\in (0,1) $$ Force $y=0$(suppose $0$ is in the domain of $f(x)$), we ...
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2answers
26 views

If we delete two points $x,y$ from $\mathbb{A}^1$, can we without loss of generality assume $x=0, y=1$?

My intuition is that we can assume this. More precisely, what I mean is, suppose $\mathbb{A}^1_k$ is the affine space over an algebraically closed field $k$. If $x,y$ are any two distinct points in ...
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22 views

What are the prerequisites to understand Affine Invariant Fourier Descriptors?

I need to implement Affine Invariant Fourier Descriptors on matlab, the objective is to compare two objects one reference and other transformed by affine transformation for recognition, my problem is ...
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2answers
72 views
+100

Why is $\text{Mor}_{\mathrm{reg}}(*,W)\to \text{Hom}_{k-\mathrm{alg}}(k[W],k[*])$ not surjective when $W=\mathbb{A}^2\setminus\{(0,0)\}$.

Suppose we're working over an algebraically closed field $k$. If $V\subseteq\mathbb{A}^n$ and $W\subseteq\mathbb{A}^m$ are affine algebraic sets, there is a well known bijective correspondence $$ ...
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1answer
53 views

Why is $f(X)$ open or closed if $f:X\to\mathbb{A}^1(k)$ is regular?

I have a question about a certain property of regular maps into $\mathbb{A}^1(k)$. This is my notation for the affine space over $k$, algebraically closed. Suppose $f:X\to \mathbb{A}^n(k)$ is a ...
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19 views

Computing the dimension of the affine hull of a set of vectors

I have a few hundred vectors that live in $R^d$ with $d$ larger than 100. I'm interested in finding the dimension of the affine hull of these points. Computing the dimension of the linear subspace ...
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1answer
37 views

If $n\geq 2$, why is $k[\mathbb{A}^n\setminus\{p\}]=k[\mathbb{A}^n]$?

Assuming $n\geq 2$, why is the coordinate ring of affine $n$-space over an algebraically closed field $k$ unchanged if we delete a point? That is, if $p\in \mathbb{A}^n$, why does ...
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1answer
22 views

how to obtain transformation matrix A in y = Ax + b notation?

I'm trying to obtain original transform matrix A and its translation vector b From y=Ax+b equation. I have original values of vectors before transform and translation (x) and vectors after transform ...
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3answers
82 views

Why are $xy=0$ and $x^2-x=0$ not isomorphic?

I'm not sure how to rigourously back up my intuition that the curves given by $x^2-x=0$ and $xy=0$ in $\mathbb{A}^2_k$ are not isomorphic. If I denote the varieties as $V_1=V(xy)$ and ...
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0answers
19 views

Discrete Geometry (Polytopes)

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset R^d$ be a point configuration affinely spanning $R^d$ (i.e., $aff(V) = R^d)$. Let H be the collection of hyperplanes spanned by ...
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1answer
32 views

Why is every open in $\mathbb{A}^1$ necessarily principal?

Let $U\subseteq\mathbb{A}^1$ be an open set in affine $1$-space. Why is $U$ necessarily a principal open set? Since $U$ is the complement of a closed set, I write $U=\mathbb{A}^1\setminus V(S)$ for ...
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1answer
25 views

Express a point as an affine combination of another two points(3D collinearity)

So, given the points A(1,2,2), B(2,4,2) and C(3,6,2) I have to show that they are collinear. ...
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1answer
31 views

Finding a line $L\subset V(y-xz)\subset\mathbb A^3_k$

I want to find lines $L\subset V(y-xz)$ and $M\subset\mathbb A_k^2$ such that $$ V(y-xz)\setminus L \simeq \mathbb A_k^2\setminus M\ . $$ Hint suggests that I use the projection $(x,y,z)\mapsto(x,y)$. ...
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1answer
27 views

Are quasiaffine subsets of $\mathbb{A}_F^n$ always necessarily open or closed?

Something I was wondering about lately, suppose $\mathbb{A}_F^n$ is affine space over a field $F$ which is algebraically closed. When I say a quasi-affine set, I mean a set that is locally closed, so ...
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0answers
55 views

What are sliding vectors mathematically?

What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
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1answer
22 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
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1answer
49 views

Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
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1answer
34 views

Making a matrix full rank through affine transformations

If I have (finite) $k$ vectors, $u_1,...,u_k\in\mathbb{R}^N$ that are in general linearly dependent is it possible to take positive affine transformations of the form: $$u'_i=\alpha_i u_i ...
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1answer
32 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
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1answer
24 views

Application of Desargues' theorem for constructions

I found this interesting document (german) on the internet. On page 8 it says: "Draw a line segment between two given points only using compass and ruler, while the distance between the two points is ...
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1answer
32 views

Prove that the set of all fixed points is a hyperplane

Let $A$ be a $n$-dimensional affine space ($ 2 \leq n$) and let $\Phi:A\to A$ be a bijective affine mapping, which isn't the identity mapping, with the following property: For all points $p$ and $q$ ...
2
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1answer
47 views

Decomposition of shear matrix into rotation & scaling

How can I decompose the affine transformation: $$ \begin{bmatrix}1&\text{shear}_x\\\text{shear}_y&1\end{bmatrix}$$ into rotation and scaling primitives? $$ ...
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1answer
42 views

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: ...
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1answer
23 views

How to orthogonalize a set of 2x2 matrices?

I have set of 2D affine transformations of images and I need to modify the transformations such way that they become as close to rotations as possible to minimize distortions of images. Let the ...
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1answer
26 views

The set if affine?

The Set $\{ Ax + b | Fx = g \}$, is it affine? How can I prove it? My answer is yes, the intuition is that $\{ x | Fx = g \}$ is a solution space of equation $Fx = g$, thus it is a linear subspace. ...
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22 views

When the sum of coefficients of two linear combinations are equal.

I recently was looking a set of polynomials (the Legendre polynomials up to degree $n$) that form a basis for the space of polynomials $\{a_{0} + a_{1}x + \dots + a_{n}x^{n}: a_{i} \in \mathbb{R}\}$ ...
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1answer
46 views

Affine connection

The affine connection is not in general defined uniquely by the smooth structure and the Riemannian metric. Can you give some demonstration with some examples?
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11 views

Is the below formula equivalent?

$K$ is a simplicial complex: Is $\{\sigma \in K | \sigma \cap conv(\{a, b\}) = \emptyset\}$ equivalent to $\{ \sigma \in K | \sigma \cap \{ a \} = \emptyset \} \cap \{ \sigma \in K | \sigma \cap \{ ...
2
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1answer
37 views

Difference between Euclidean space and $\mathbb R^3$

What is the difference between Euclidean space and $\mathbb R^3$? I have found in some books that they are the same, but in other references like Wikipedia, it says that a vector in $\mathbb R^3$ is ...
2
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0answers
27 views

Linearizable reductive group action.

Let $k$ be a 0 characteristic field, and $G$ a reductive group in $GA_2(k)$. How is it possible to deduce that $G$ is conjugated to a subgroup of $GL_2(k)$ ?
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2answers
31 views

affine hull, how to understand the statements below?

I am new to affine space, I looked through the wikipedia page, and have problem understanding the statements below. The affine hull of a set of three points not on one line is the plane going ...
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0answers
11 views

Affine transformation step by step

So I have the before and after 3d coordinates of an object that has been translated and rotated. I need to calculate the matrix to return the object to it's original position. I've been reading ...
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18 views

V is an affine subspace iff for any two distinct points V contains the line dtermined by these points

Since it can be shown that the barycenter of n weighted points can be obtained by repeated computations of barycenters of two weighted points, a nonempty subset V of E is an affine subspace iff for ...
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8 views

the set $V$ of barycenters ${\Sigma}_{i\in I}{\lambda}_ia_i$ is the smallest affine subspace containing $(a_i)_{i\in I}$

Given an affine space $(E,E^{\to})$, for any family ${(a_i)}_{i\in I}$ of points in E, the set V of barycenters ${\Sigma}_{i\in I}{\lambda}_ia_i$ is the smallest affine subspace containing ...
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1answer
15 views

Let $a,b$ be affine combinations of points from a set $S$. Then is the affine combination of $a,b$ also an affine combination of points from $S$?

Let A be an affine space, $a,b$ affine combinations of points from a finite subset $S$ of A. Then is the affine combination of $a,b$ also an affine combination of points from $S$? I found it ...
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2answers
75 views

Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
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30 views

An easy definition of an $n$-dimensional affine cube

In a few weeks I'm giving a presentation on the History of Ramsey Theory and I want to start off with Hilbert's cube lemma. The only problem is that the pre-requisites for this course is only ...
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1answer
20 views

Alternative characterization of a finite dimensional affine set

As the definition in the S. Boyd's textbook: My question is the following representation: What is the relationship between this representation and the definition above it? EX: sum of elements ...
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5answers
1k views

What are differences between affine space and vector space?

I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by ...
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1answer
19 views

Transform gradient to reference element

Minimal example of the problem How can you transform the gradient to the reference element?
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30 views

Dumb question regarding affine transformations

So, we can write an affinity $\phi$ as $$\phi(x) = Ax + b$$ for some linear transformation $A$ and vector $b$. What exactly does it mean for an affinity to be a "scaling"? Is this a mapping of the ...
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1answer
21 views

Prove that this affinity is the identity mapping

Let $\phi : K^n \to K^n$ be an affinity, such that all lines are parallel to their image under $\phi$. Prove that if $\phi$ has two fixed points, then $\phi$ is the identity mapping. My attempt: I ...
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Interesting observation WRT 1,2,3-dimensional convex polytopes and higher dimensional ones as counterpart

When I experimenting with qhull utility and Quickhull Algorithm, I found that in $\mathbb{R}^d, d \in \{1,2,3\}$ space the number $F$ of convex hull's $(d - ...