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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1answer
68 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 ...
1
vote
1answer
44 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 ...
0
votes
1answer
55 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 ...
4
votes
3answers
102 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 ...
1
vote
1answer
45 views

The number of $(d-1)$-faces in a $d$-polytope is at least $(d+1)$

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset \mathbb{R}^d$ be a point configuration affinely spanning $\mathbb{R}$ (i.e., $\operatorname{aff}(V) = \mathbb{R}^d)$. Let $H$ be ...
2
votes
1answer
39 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 ...
0
votes
1answer
68 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. ...
0
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1answer
37 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 $(x,y,z)\mapsto(x,y)$. ...
0
votes
1answer
46 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 ...
0
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0answers
391 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 ...
1
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1answer
52 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 ...
1
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1answer
60 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 ...
0
votes
1answer
55 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 ...
2
votes
1answer
90 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 ...
1
vote
1answer
61 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 ...
0
votes
1answer
40 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
votes
1answer
134 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? $$ ...
0
votes
1answer
77 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: ...
2
votes
1answer
30 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 ...
0
votes
1answer
31 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. ...
0
votes
0answers
57 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 $\{a_{0} + a_{1}x + \dots + a_{n}x^{n}: a_{i} \in \mathbb{R}\}$ ...
0
votes
1answer
139 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?
3
votes
1answer
70 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 ...
1
vote
0answers
35 views

Linearizable reductive group action.

Let $k$ be a 0 characteristic field, and $G$ a reductive group in $GA_2(k)$ (the group of automorphisms of k[x,y] as k-algebra). How is it possible to deduce that $G$ is conjugated to a subgroup of ...
0
votes
1answer
93 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 ...
0
votes
1answer
19 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 ...
1
vote
2answers
120 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?
1
vote
1answer
22 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 ...
18
votes
6answers
4k 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 ...
0
votes
1answer
29 views

Transform gradient to reference element

Minimal example of the problem How can you transform the gradient to the reference element?
0
votes
0answers
48 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 ...
1
vote
1answer
27 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 ...
1
vote
0answers
54 views

A confusion regarding Affine spaces

Take an Affine space $\Bbb{A}$ over the field $K$. How would you determine the points satisfying any polynomial $f(x)$? If there is no fixed origin, points can be given names with reference to ANY ...
0
votes
2answers
46 views

Number of parameters to specify an affine transformation in n dimensions

In general, how many parameters does it take to specify an affine transformation in $n$ dimensions, and how does one go about proving this? For example, in 2 dimensions it takes 6 parameters, and in ...
0
votes
1answer
96 views

Formula of signed distance from hyperplane to point

Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$? I ...
1
vote
1answer
39 views

linearization of $\log(|x|)$

I am trying to convexify $\log(|x|)$. I think its concave. So I am trying to get an affine upper bound through linearization. But the problem is there are two concave functions because of the absolute ...
2
votes
1answer
54 views

Is a similarity map necessarily affine linear?

My text on fractal geometry introduces the following definition: A map $S: \mathbb R^n \to \mathbb R^n$ is called a similarity map if $$\exists c>0 \ \forall x,y \in \mathbb R^n: ...
2
votes
1answer
67 views

Computing the Zariski cotangent space

I'm an extreme beginner with algebraic geometry and am trying to get used to things. Say I have some (algebraically closed) field $k$, in $k^2$ I want to compute the Zariski cotangent space, let's say ...
0
votes
0answers
88 views

Why are affine subspaces also sometimes called linear manifolds?

According to Wikipedia, an affine subspace is a subset of a vector space closed under affine linear combinations. That is, linear combinations whose scalar coefficients sum to 1. It's not clear to ...
0
votes
1answer
109 views

Affine Bundles vs Affine Spaces

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. ...
1
vote
1answer
32 views

Angle of two planes in $\mathbb E_4$

I have two planes (given in parametric form) in $\mathbb E_4$: $\alpha$: (7,3,5,1) + t(0,0,1,0) + s(3,3,0,1) and $\beta$: (1,5,4,1) + r(0,0,0,-1) + p(2,0,0,1), and I have to find angle between them. I ...
0
votes
1answer
23 views

Newbie with barycentric coordinates: why one is zero when on a vertex?

I'm trying to calculate if a 2D point lies inside a triangle and I solved the following system: ...
2
votes
1answer
96 views

intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at ...
3
votes
1answer
86 views

local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ ...
0
votes
2answers
60 views

intersection multiplicity at non-zero point

Compute the intersection multiplicity of $f=x+y-2$ and $g=x^2+y^2-2$ at $(1,1)$. Is this the same as the intersection multiplicity of $f(x+1)$ and $g(x+1)$ at $(0,0)$ which I have computed to be 2? ...
3
votes
2answers
132 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or ...
1
vote
1answer
90 views

Find intersection multiplicities

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and ...
0
votes
1answer
38 views

Non constant function of two points invariant under Affine transformation proof

Here is the question; Prove that there does not exist any nonconstant function of pairs of distinct points $P,Q\in\mathbb{R}^2$ or of triples of distinct non collinear points $P,Q,R\in\mathbb{R}^2$ ...
2
votes
0answers
77 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
2
votes
0answers
52 views

A question for epigraph and affine function

I'm working with a problem the epigraph of a real-valued function $f$ is a halfspace $\iff$ $f$ is a real-valued affine fuction. First, I quickly recall some definitions: A (closed) halfspace is a ...