0
votes
1answer
27 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
33 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
0answers
10 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
0answers
7 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 ...
8
votes
5answers
595 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
2answers
16 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 ...
2
votes
1answer
37 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: ...
1
vote
1answer
30 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
16 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: ...
0
votes
0answers
50 views

5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
-2
votes
1answer
47 views

Finding the number of solutions to two equations

I have a question: For the following system of linear equations, using Gaussian elimination, decide whether it has at least one solution. If it does, represent the general solution as an affine map ...
0
votes
1answer
132 views

Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
0
votes
1answer
32 views

Relations between an affine space and a topological space

What is the relation between an affine space and a topological space? Is one a specialization of the other? Moreover, what do we call a point in geometry: an element of a topological space or an ...
0
votes
3answers
110 views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
1
vote
1answer
205 views

Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

I am trying to understand the exact relation between all these things: point vector affine space vector space basis frame coordinate system Can you explain me rigorously (in the mathematical ...
0
votes
1answer
69 views

Show existence of linear transformation from subset to subspace embedded in $\mathbb{F}_2^n$

Assume I have a subset $X$ (not necessarily a subspace) of $\mathbb{F}_2^n$, of size $\leq 2^{n-1}$. It seems likely to me that there should exist a bijective linear transformation taking $X$ to a ...
0
votes
1answer
29 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
0
votes
1answer
36 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 ...
1
vote
0answers
51 views

Linear algebra, affine space, and floor function

My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for ...
2
votes
0answers
30 views

Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
0
votes
1answer
43 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
0
votes
1answer
154 views

Affine transformation matrix coefficients

In an affine transformation $x \mapsto Ax+b$, $b$ represents the translation; but what does the matrix $A$ represent exactly? In a 2D example, $A$ is a $2\times 2$ matrix, but what does each term ...
0
votes
0answers
24 views

Expressing plane and line as affine subspaces

I need to express the following plane, line and the intersection of the two as affine subspaces with a vector $v \in \mathbb{R}^3$ and a subspace $W \subseteq \mathbb{R}^3$. $$P = \{(x_1, x_2, x_3) ...
1
vote
1answer
395 views

Finding an affine combination of a point on a triangle

I have a problem involving affine combinations that I can't figure out how to solve. Given the above picture, write q as an affine combination of u and w. Now, I understand how to write the ...
0
votes
1answer
65 views

alternative definition of Affine map

Let $f:X\longrightarrow Y$ be a function on real vector spaces (note that $X,Y$ have arbitrary dimensions). If $T(x)=f(x)-f(0)$ is linear, $f$ is called an affine map. Prove that $f$ is affine if ...
1
vote
1answer
133 views

Affine group, semi-direct product and linear transformations

According to wikipedia the Affine group is the semi-direct product of a vector space $V$ and the general linear group $GL(V)$. Here is the definition of the semi-direct product in terms of matrices ...
4
votes
1answer
85 views

Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$

I don't know how to prove the following: Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$, where $A \in GL_2(\mathbb{R})$, $b$ is a fixed vector in ...
3
votes
1answer
123 views

Find the tangent space of $\mathrm{Aff}(n)$

Find the tangent space of $\mathrm{Aff}(n)$. see Proof: Tangent space of the general linear group is the set of all squared matrices $\mathrm{Aff}(n)$ is the set of all matrices of the form $$ ...
0
votes
1answer
35 views

Effect of Moving within the Feasible Region

$f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a concave function with local maximum at $\mathbf{x}^*$ in a convex, closed feasible set $\mathcal{F}\subset\mathbb{R}^n$. Now consider a suboptimal point ...
1
vote
0answers
104 views

Number of vector and affine subspaces of dimension $ k$ of $E$ over $\mathbb{F_q}$

Problem (comments after): Let $\mathbb{F_q}$ be a finite field of cardinal $q$ and $\mathcal{E}$ an affine espace of dimension $n$ directed by the vector space $E$. Show that: ...
1
vote
1answer
185 views

Line-preserving transformations

Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection $p$ in ...
1
vote
1answer
140 views

Using absolute coordinates in 2D affine transformation matrix

In my 2D animation program I have a sprite which transformation is described by a 2D affine transformation matrix (SVGMatrix): $$ \begin{bmatrix} a & c & e \\ b & ...
2
votes
1answer
80 views

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 ...
0
votes
0answers
68 views

Multi-affine function

Suppose i have a three-variable function f(x1,x2,x3), f:R^3 -> R. If it is linear for x1,x2 and x3 we can say it has the form f(x1,x2,x3) = c1x1 + c2x2 + c3x3 where c1,c2,c3 in R. We can ...
1
vote
1answer
654 views

Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?

I'm a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a square into a quadrilateral (as ...
0
votes
1answer
43 views

Affine transformation, if $L_1, L_2 - $ skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective

Could you tell me how to prove that if $f$ is affine transformation, $L_1, L_2 $ are skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective?
1
vote
0answers
80 views

Distance between two affine lines using determinant of Gramian matrix.

I've a task to find the distance in $E^4$ between: $L = [1,2,-1,4] + \text{lin}((1,2,-1,0))$ and $M = [2,3,1,5] + \text{lin}((2,1,0,2))$ My efforts to find the correct solution: Let ...
2
votes
0answers
50 views

$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$

Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$ Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis. I don't ...
2
votes
1answer
143 views

On affine spaces, distances, angles, and coordinates

Let's define an affine space as a pair $(A, V)$, where $A$ is a set and $V$ is a vector space, together with a map $V\times A \rightarrow A, \;\; (v, a) \mapsto v + a,$ such that $\forall \, a \in ...
2
votes
1answer
111 views

How to find equation system describing affine space, having base of linear space and a vector

How to find equation system describing affine space, having base of linear 'overspace' and a vector? Suppose that I've vectors $\alpha$ and $\beta$, so that $W=\text{lin}(\alpha, \beta)$, and a ...
2
votes
2answers
87 views

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$ So I've tried to put the equation of plane ...
1
vote
4answers
91 views

Find the equation of plane containing line described by

Please help me in this really easy task Find the equation of plane containing line described by $x+3y-2z=1$, $2x-y+2z=3$, containing point $(1,1,3)$
1
vote
2answers
157 views

Equation of plane — point/vector pedagogy

Suppose we have a point $\mathbf P$ and a vector $\mathbf n$ in plain ordinary 3D space. Here I am deliberately using upper-case letters for points, and lower-case points for vectors, since they are ...
5
votes
3answers
481 views

Showing that if $fg=gf$ and $fh=hf$, then $gh=hg$, where $f$, $g$, and $h$ are affine functions

Given real numbers $a$ and $b$ ($a \ne 0$), let $f_{a,b}$ be the function $\mathbb{R} \to \mathbb{R}$ defined by $x \mapsto ax+b$. The set of such functions is a permutation group on $\mathbb{R}$, ...
3
votes
1answer
500 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
293 views

Intersection of affine subspaces is affine

So if I have two affine subspaces, each is a translate ( or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is again affine, particularly in ...
0
votes
3answers
177 views

Equation of a line in homogenous coordinates given 2 points in affine coordinates

So if I have 2 points $A$ and $B$ such that $F(A) = (1; a, a^3)$, and $F(B) = (1; b, b^3)$. how do I find the equation of this line in homogeneous coordinates? So I know how to get a line the ...
3
votes
1answer
243 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 ...
3
votes
1answer
121 views

Cross-ratio relations

The way I define the cross-ratio in projectve geometry: Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
0
votes
1answer
112 views

Cross-ratio projective geometry

I have 4 points $P_0=[1:2], P_1=[3:4], P_2=[5:6], P_3=[7,8]$ in $\mathbb KP^1$ and would like to evaluate the cross-ratio. It is given by the following: $\pi:\mathbb KP^1\rightarrow G$ is the unique ...