Questions tagged [projection]

This tag is for questions relating to "Projection", which is nothing but the shadow cast by an object. An everyday example of a projection is the casting of shadows onto a plane. Projection has many application in various areas of Mathematics (such as Euclidean geometry, linear algebra, topology, category theory, set theory etc.) as well as Physics.

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Galerkin Methods

Let $(\varphi _n)_{n\in \mathbb{N}}$ be a basis of an Hilbert space $H$. Define $H_N=\operatorname{span}\{\varphi _1,\ldots ,\varphi _N\}$. Suppose that there exits $a:H\times H \to \mathbb{R}$, a ...
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Euclidean projection on convex set of positive semidefinite matrices

Define the Euclidean projection for a convex set $C$ as follows $$\pi_C(y) := \min_{x \in C} \| y - x \|_2^2$$ How would we find the projection map when $C$ is the cone of positive semidefinite ...
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Projection of a vector on a span

I have the vectors: u=$(-2,-2,-2)$, v=$(3,-1,2)$ I have to find a vector - w=$(x,y,z)$ ( I think ) on v and u. projection of w on v is $-6v$ projection of w on u is $6u$. I got to that point: $-6|v|^2=...
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Find location and orientation of a pinhole camera from a given image of a triangle with known sides contained in a known plane

So, I am playing with problems related to perspective images produces by a simple pinhole camera. I came up with the following problem. Suppose you have a triangle of known side lengths, that is ...
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Express a normal linear transformation as a linear combination of projections.

There's the following theorem: Let $T:V\rightarrow V$ be a normal linear transformation $(TT^{*}=T^{*}T)$. Prove there exists $E_1,...,E_k$ such that: $E_i^{2}=E_i$ $E_iE_j=0$ for all $i\neq j$ $\...
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Optimal (in terms of remaining vector lengths) 2-dimensional projection plane of $n$ $d$-dimensional unit vectors

I have a finite number of $n$ unit vectors in $\mathbb{R}^{d}$. I would like to find a two-dimensional projection plane such that each vector has a length larger than 0 in the projection. Moreover, I ...
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Proving a formula about inner product of a vector and its projection onto a orthogonal complement of the column space

We have $h_1,...,h_{i},...,h_K \in \mathbb{C}^N, N\gt K$ and they are linearly independent. Then we define $v \triangleq \prod^{\bot}_{[h1,...,h_{i-1}\ ,\ h_{i+1} \ ,...,h_K]}h_i $, where $\prod^{\bot}...
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Trying to understand projection onto a plane, subspace of R^3, of a point b when there is no solution for Ax = b

I was following along some online classes for linear algebra and was doing fine until it came to projections onto subspaces and I really really could not understand the proof and logic behind the ...
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Is the following statement in group theory true?

Let $G$ be a group such that $$G=\prod G_i,$$ where above product is arbitrary. Suppose that there exists a subgroup $H$ of $G$ such that $$\prod G_i=H\prod G_i',$$ where $G_i'$ is derived subgroup of ...
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Project 'b' onto 'a' to find out the projection of 'b' onto 'a'

[Question: Project 'b' onto 'a' to find out the projection of 'b' onto 'a'. To solve this, which formula should be used to find out the projection? Scalar Projection or Vector Projection? Lil bit ...
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Projection mapping as quadratic programming problem.

I am studying Variational inequalities in Hilbert space. To define this, let $C$ be a closed, convex subset of a Hilbert space $H$. The Variational inequality problem is to find $x \in C$ such that $\...
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projection of a non-zero mean Gaussian vector into a Ball

Let $d$ denote the dimension, $\mathbf{B}_d$ denote the ball of radius one in $\mathbb{R}^d$. For $x\in \mathbb{R}^d$ let $\Pi_{\mathbf{B}_d}(x) = \frac{x}{\max\{1,\|x\|_2\}}$. Consider a fixed vector ...
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projection of the data along the 1st k principal components

I'm a final year maths undergrad doing a course in multivariate data analysis, but I'm really struggling with the linear algebra. In particular the “projection of the data along the 1st k principal ...
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What is a symplectic structure on a smooth vector bundle

We just had the definition of a symplectic structure on a vector bundle in the lecture and I am having trouble understanding it Definition: Let $\pi : E \to M$ be a smooth vector bundle. Then a ...
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How to calculate the mass of a 3-D sphere collapsed into 2-D plane?

My sphere has density, $p(R)=(R+10)^-2$. If I collapse this sphere into a $2D$ plane, let's say it forms a $2D$ ellipse as a result, how will I calculate the mass of this $2D$ ellipse? in what ...
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Projecting a point onto a convex set given by Log-Sum-Exp

Motivated by a wish to encode signal temporal logic specs (with linear predicates) as optimization problems w/o mixed integer approaches, I've been attempting to find a way to define the projection ...
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Orthogonal projection of an ellipsoïd from N to 2 dimensional space

Suppose we have a $N\times N$ symmetric-positive-definite matrix $A$, representing an ellipsoïd in $N$ dimensional space. How to find the matrix $A_{xy}$ corresponding to orthogonal projection of ...
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Projection of a matrix with respect to a certain norm

Suppose I have the matrices $M_0$ and $M$. I need to find the projection of $M$ onto a $\epsilon$ sphere generated by $M_0$ i.e., given $M_0$ and $M$ find $\tilde{M}$ such that $||\tilde{M} - M_0|| = \...
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Projection of tensor in vectorial space

Let us consider the closed convex cone of Sym (Sym be the subspaces of Lin constituted by all symmetric tensors) K={A∈ Sym such that trA ≤ 0 } For each E∈ Sym determine the projection P(E) of E onto K
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Find a point of the plane which distance to a line minimal is

Let be $P \subseteq R^3$ (P is a plane)and $X_1, X_2 \subseteq R^3$ two lines given by: $P = \Bigg\langle \begin{pmatrix} 2 \\ 0 \\ 1\end{pmatrix},\begin{pmatrix} 0 \\ 2 \\ 1\end{pmatrix} \Bigg\...
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Existence of well defined map $R/J\to R/I$ implies $J\cong M\subseteq I$?

I'm currently trying to prove by myself some proposition related to Hopf-Galois theory (from the paper "Galois Correspondences for Hopf Bigalois Extensions", by Peter Schauenburg). The ...
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Notation for inner product between scalar-valued and vector-valued functions

Consider $\mathbf{f}:\mathcal{S}\rightarrow\mathbb{R}^{n}$ and $\left(\phi_k\right)_{k=1}^{\infty}$, where $\phi:\mathcal{S}\rightarrow\mathbb{C}$, and $\mathcal{S}\subset\mathbb{R}^{n}$. My intent is ...
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projection of band limit functions

Define $$ P_{\Omega}f(x) = \frac{1}{2\pi}\int_{\mathbb{R}}e^{i\zeta x}\chi_{\{|\zeta|\leq \Omega\}}\hat{f}(\zeta)d\zeta $$ where $\hat{f}$ is the Fourier transform of $f$. Show that $P_{\Omega}^2 = P_{...
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Flattening and projection of a covariance matrix onto a vector in the x-y plane

I am working with a covariance matrix $P \in R^3$ that I need to simultaneously project onto a vector $\mathbf{v}_f \in R^3$ and flatten into the plane $z=0$. A paper in my posession states that the ...
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1 vote
1 answer
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Orthogonal project of a vector $\vec{y}$ that sit in the same plane (aka span($u_1, u_2$)) such that $\vec{y},\vec{u_1}, \vec{u_2} \in \mathbb{R^3}$

Confirmed that $u_1,u_2$ are indeed orthogonal by taking their dot product which equals zero. Correct answer = $\begin{pmatrix}-1\\ -1\\ 6\end{pmatrix}$. Geographically, mapped out in geogebra: we see ...
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Evaluate the projection operator norm with respect to maximum norm

Consider the normed space $\left(\mathbb{R}^{3},\|\cdot\|\right)$ where $\|\cdot\|=|\cdot|_{\infty}$ is the usual maximum norm. Consider the 2 -dimensional vector subspace $\left \{ (x,y,z):x+y+z=0 \...
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Spectrum of general projection and orthogonal projection

I am trying to think about this, but I seem to be stuck. Suppose $P$ is a projection on a Hilbert space $\mathcal{H}$. If I am just talking about a general projection, where I only know that $P^2=P$, ...
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Looking for an example for the projection's theorem on an inner product space?

I'm looking for an example of a non-empty, non-convex and complete subset $C$ of an incomplete inner product space $E$ such that if we apply the projection's theorem on $C$ it gives several (maybe ...
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Calibrating a pinhole camera (finding $z_0$)

A pinhole camera is a very simple theoretical device for generating perspective images on a plane that a distance $z_0$ from the pinhole (a point) and whose normal vector is the direction vector at ...
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Locating vertices of a known triangle in $3D$ from a single image

Suppose you have a labelled triangle with known side lengths, and you take one image of this triangle using a known pinhole camera (i.e. the focal length is known), from a point with known coordinates,...
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Locating a point in $3D$ using two perspective images

Suppose I am given a point $P(x, y, z)$ where $x, y, z$ are unknown. I take two images (perspective projections) of this point using a simple pinhole camera with an unknown focal length $f$, from two ...
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3 votes
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Are the projections of a direct sum continuous?

Let $(V, \lVert \cdot \rVert)$ be a normed vector space, and let $X, Y \subseteq V$ be linear subspaces such that $X + Y$ is a direct sum (that is, $X \cap Y = \{0\}$). Since the sum is direct, every ...
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Numerically approximating projection onto an infinite-dimensional Hilbert-space

We have the following problem that we want to model numerically. We would be glad for any references, since we could not find much useful information on these kind of problems and since we do not come ...
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Does this continuous function admit a projection on this set?

Let $V:=(\mathcal{C}([0,1], \mathbb{R}),\Vert .\Vert)$ the vector space of the continuous functions on $[0,1]$ endowed with the norm $\Vert . \Vert$ which satisfies for all $f \in V : \Vert f \Vert = \...
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Why is the orthogonal complement actually orthogonal to the projection?

How do you show that $q=x-p$ is orthogonal to the projection ($p$)? I can understand it intuitively from drawing it out. But can't show it's true with proof. What I know is we will have to show that ...
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Show whether the scalar projection is idempotent

A while ago I asked Show whether the vector projection is idempotent, which made it clear to me that the idempotency of the vector projection could be checked with a simple substitution. Now I am ...
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quotient topology on $A^{*}$ equals subspace topology on $\pi(A)$ for $A^{*}:= \pi^{-1}(\pi(A))$ open or closed.

Let $\pi:X \to X/\sim$ be a quotient map. Let $A \subset X$ and define $A^{*} := \pi^{-1}(\pi(A)) \subset X$. If $A^{*}$ is open or closed in X, then the subspace topology on $\pi(A)$ is the same as ...
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Brezis's Ex 3.32: projection on the domain of a proper convex l.s.c. map

I'm doing Ex 3.32.(5. and 6.) in Brezis's book of Functional Analysis. Could you have a check on my attempt? Let $(E, |\cdot|)$ be a uniformly convex Banach space and $C \subset E$ a nonempty. Prove ...
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Distance between two closed convex bounded non-empty subsets in a CAT(0) space

Let $(X,d)$ be a complete CAT(0) space, where $d$ is the metric on the space $X$. Being CAT(0) here means that for any $x,y\in X$, there is some $m\in X$ such that for any $z\in X$, we have the ...
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2 votes
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Projection is uniformly continuous on bounded sets of a uniformly convex Banach space

I'm doing questions 2., 3., and 4. of Ex 3.32 in Brezis's book of Functional Analysis. Let $(E, |\cdot|)$ be a Banach space and $C \subset E$ a nonempty closed convex set. Assume that $E$ is ...
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Proving Theorem 10.3 on Steven Roman's Advanced Linear Algebra

I want to prove the (3) and (4) of theorem which says $\textrm{im}(\tau \tau ^{*})=\textrm{im}(\tau )$ and $(\rho _{\textrm{S,T}})^{*}=\rho _{\textrm{T}^{\perp },\textrm{S}^{\perp }}$, here $\tau ^{*}$...
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generalization oh lhuillier´s thorem

lhuillier´s thorem states thatEvery triangle can be considered as the normal projection of a triangle of given form (an equilateral triangle, for indtance). see the discussion here : Projection of an ...
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1 vote
2 answers
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Projector of a euclidean space

I have a Euclidean Space $V=R^3$ and a subspace E = {v} , v≠0 so to calculate the projection of $u \in V$. My teacher wrote on the blackboard: P(u)=$\frac{<u,v>}{||v^2||}v$ so for this case ...
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Finding adjoint and norm of projection

We consider the Hilbert space with standard inner product, $0<\phi<\frac{1}{2} \pi$. We consider the projection P: $ P \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} x_1-x_2cot(\phi)\\...
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Proof that dual of $L^p$ is $L^q$

Im working through a proof which shows that for $\frac{1}{p}+\frac{1}{q}$ the map $\Phi:L^q(\Omega;\mathbb{K})\to (L^p(\Omega;\mathbb{K}))^*$, $g\to \Phi(g)[f]:=\int_{\Omega}\overline{g}f\space d\mu$ ...
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Signed Distance of a Point to a Hyperplane

Let's define a affine function $f(x) = \alpha^Tx+\alpha_0$ where $\alpha$ is a vector (weights) and $\alpha_0$ is a constant (bias). Also, define a normal vector of the hyperplane as $n=\frac{\alpha}{...
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Is the dot product a projection?

Define $$\boldsymbol{\mu}=\frac{1}{N}\sum^N_{i=1}\boldsymbol{x_i}$$ If we have a direction vector $\boldsymbol{w}$, The book stated that the projection of the mean onto the line $\boldsymbol{w}$ is ...
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Projection onto a dilated set

Let $C \subset \mathbb{R}^n$ be non-empty, closed and convex. Also let $\delta>0$ and consider the dilated set defined as follows $$ D_{\delta}(C) = \{ x \in \mathbb{R}^n: \| x - \pi_{C}(x)\|_2 \le ...
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2 votes
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Gradient of a function involving the Euclidean projection

Given a non-empty, closed and convex set $C \subset \mathbb{R}^n$, the Euclidean projection of $y \in \mathbb{R}^n$ onto $C$ is given by $$ \pi_C (y) = \arg \min_{x \in C} \frac12 \| y-x\|_2^2. $$ I ...
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Projection of Torus in 2D

Is there a polynomial expression for the contour of the projection of a 3D torus in 2D? I have difficulties with the concept of projections. As far as I understand the question we map the torus on ...
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