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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16 views

Efficient projection orthogonal to multiple vectors

Suppose I have $M$ vectors $a_1, a_2, ..., a_M$ in $\mathbb R^N$, with $M < N$. Given another vector $x\in\mathbb R^N$, I want to find the projection of $x$ in the sub-space orthogonal to all the $...
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Approximation of projection of a function and computation of dual basis

Let $H=L^2(0,1)$ and $M$ be a linear manifold generated by $n$ independent functions, $\{b_1, \cdots, b_n\}$. Define the orthogonal projection $P_M : H \rightarrow M$ on $H$ into $M$. Then, we have ...
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1answer
17 views

Compute projection of vector onto nullspace of vector span

Say I have a matrix $\pmb{W}$ of $m$ vectors, each of length $n$: $\pmb{W} =\left[ \vec{W}_1, \dots, \vec{W}_m\right]$, where $\vec{W}_i \in \mathbb{R}^n$ for integers $1\leq i\leq m$. How would I go ...
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What is the value of 2a? [closed]

Consider $W=\operatorname{span}\left(\begin{bmatrix}1\\1\end{bmatrix}\right)$. Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ be a matrix such that $\operatorname{proj}_W(v)=Av$ for all $v\in\...
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is linear projection sufficient for capturing all extreme points?

Given a set $X \subset R^n$ with $m$ points. We can find it's Convex Hull and together with set of extreme points $E(X)$. And none of any points are linear multiplier of each other. Under a linear ...
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+50

Recovering three dimensional vectors after projection and cross product

Suppose $e_i \in \mathbb{R}^3$, $1\leq i \leq 3$ with $\Vert e_i \Vert=1$. Suppose $u,v \in \mathbb{R}^3$, $u^T v=0$, $e_i^T u \neq 0$, $\Vert u \Vert =1$. Suppose $k\in \mathbb{R}$. Define the ...
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1answer
30 views

Proving a mathematical statement with projections

Let $a, x \in \mathbb{R}$, $S \subset \mathbb{R}^n$ a convex and closed set and $L = S + \{x\}$. Prove that $L$ is convex and closed and that $proj_L(a) = x + proj_S(a - x)$. Guys, im stuck with this ...
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Projecting a vector onto a hypercube

Assume we have a vector $λ=[λ_1,λ_2,\cdots,λ_n]^Τ\in\mathbb R^n$. There is a hypercube $[\underline Λ,\overline Λ]$ where $\underline Λ=[\underline Λ_1,\underline Λ_2,\cdots,\underline Λ_n]^T\in\...
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1answer
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Eigenvector problem with ellipsoids (maximizing quadratic form)

Let $B$ be a symmetric, positive definite matrix and consider the problem $$\begin{array}{ll} \text{maximize} & x^\top B x\\ \text{subject to} & \|x\| = 1\\ & b^\top x = 0\end{array}$$ for ...
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1answer
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Rudin's Real and Complex Analysis, Section 9.16

In Section of 9.16 from Rudin's RCA, it says Let $\hat{M}$ be the image of a closed translation-invariant subspace $M \subset L^2$, nder the Fourier transfrom. Let $P$ be the orthogonal projection of ...
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Project orthonormal vectors onto subspace while preserving orthonormality

Suppose I have $m$ orthonormal vectors $u_1, ..., u_m \in \mathbb{R}^n$ where $m < n$. I would like to project each vector onto $\mathbb{R}^m$ using some linear transformation $W: \mathbb{R}^n \...
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2answers
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Cone projected tangent angle

A cone has a slope of 45 degrees. The cone is projected on a plane that is inclined to the axis of the cone by x degrees. If x was 0, the projection would be 2 lines converging at 90(45 + 45) degrees ...
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1answer
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norm and projections on inner product space

How do I show that if $\Vert Px-Qx \Vert <\Vert x \Vert$ for any $x\in V$ not $0$, then $\dim\left(M\right)=\dim\left(N\right)$. $V$ is an inner product space and $M, N$ are sub-spaces of $V$.$P$ ...
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Let $L$ be the line $y = mx$, where $m \neq 0$. Find an expression for $T(x,y)$, where $T$ is the reflection of $\textbf{R}^{2}$ about $L$.

In $\textbf{R}^{2}$, let $L$ be the line $y = mx$, where $m \neq 0$. Find an expression for $T(x,y)$, where (a) $T$ is the reflection of $\textbf{R}^{2}$ about $L$. (b) $T$ is the projection on $L$ ...
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Projection for a given embedding of von Neumann algebra.

Let $\mathcal{M}$ and $\mathcal{N}$ be two given von Neumann algebras with faithful states $\tau$ and $\eta$, respectively. Let $\varphi:(\mathcal{N},\eta)\rightarrow(\mathcal{M},\tau)$ be a $^{*}$-...
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1answer
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Equivalence of projections in smaller von Neumann algebra

I came across the following assertion and I can't understand why it's true. We are given with two finite equivalent projection $e\sim f$ in some von Neumann algebra $A$ (with a unit of course). It's ...
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a relation between the norm of the difference of projections of a vector and its norm

$V$ is an inner product space and $M, N$ are sub-spaces of $V$.$P$ is a projection on $N$ and $Q$ is a projection on $M$. How do I show that if $\left||Px-Qx\right||<\left||x\right||$ for any $x\in ...
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1answer
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Cross product and projection [closed]

$\textbf{Problem:}$ Let $v_1,...,v_{n-2} \in \mathbb{R}^n$ with $\{v_1,...,v_{n-2}\}$ is linearly independent. Let $B = (v_1,...,v_{n-2})$ and $C = \text{Col}(B)$. If $x \in \mathbb{R}^n$, show that $$...
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1answer
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Proyection of a subspace.

Let $W \subset \mathbb{R}^{4}$ a subspace generated by two vectors $$W := span \left\lbrace \begin{pmatrix} 1\\ 1\\ 0\\ 0 \end{pmatrix},\begin{pmatrix} 1\\ 1\\ 1\\ 2 \end{pmatrix} \right\rbrace. $$ ...
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1answer
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Calculate a vector that lies on plane X and results in vector b when projected onto plane Y.

I'm working on a computer program and ran into this problem that I'm struggling to figure out. Given: two planes X and Y that belong to $\mathbf{R^3}$ that both pass through the origin and are ...
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1answer
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Passage missing in simple proof of orthogonal decomposition

I am reading this nice book on linear algebra. Specifically, I am reading the proof of the Theorem for Orthogonal decomposition of a vector $x\in \mathbb{R}^n$, given a subspace $W$. I think there is ...
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Existence of a “good” projection plane

I have $n$ three dimensional vectors, all having unit length. $n$ is a positive integer. I want to find a "good" projection plane in which all vectors have a certain length. Which length can ...
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25 views

Matching set of 3D points to set of 2D points

I am working on a 3D projection problem. Basically, a user has an image of complex geometry, and a 3D object representing similar/identical 3D geometry. The software can guess relatively speaking what ...
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1answer
23 views

Projection of triangle on coordinate axes?

A triangle in the $xy$-plane is such that when projected onto the $x$-axis, $y$-axis and the line $y=x$ the results are line segments whose end points are $(1,0)$ and $(5,0)$, $(0,8)$ and $(0,13)$ and ...
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1answer
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When to apply Gram-Schmidt

Find the orthogonal projection of $(x,y,z)$ onto the subspace spanned by $(1,2,2)\ \text{and}\ (-2,2,1)$. The answer is $\frac{(5x-2y+4z,-2x+8y+2z,4x+2y+5z)}{9}$. Why is it that I do not have to ...
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1answer
23 views

Projections on Normed Spaces

Let $\mathbb{E}$ be a (real or complex) Banach space (complete normed space). Let $P$ be a projection ($P^2=P$) from $\mathbb{E}$ into itself. Is it necessarily that the norm of $P$ equals to $1$?
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1answer
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Projection Operator and Perpendicularity [closed]

$M\neq \emptyset$ including $M^\bot=\{x \in X\mid x \bot M \}$ is a vector space show that? And it is closed show that? How can i do show that?
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When is a polynomial interpolant a projection in a Hilbert space?

Let $f$ be a function in a Hilbert space, say $L^2([a,b])$. Suppose I have a degree-$n$ polynomial $p$ which interpolates $f$ at at distinct points $\{x_k\}_{0 \leq k \leq n}\subset[a,b]$. I know that ...
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19 views

Projections of a point inside the irregular tetrahedron?

I have a tetrahedron $ABCD$ and a point $P$ inside it and if $P'$ is the projection of $P$ to $\Delta ABC$ and $P''$ is the projection of $P$ to $\Delta ABD$, why do the projections of $P'$ and $P''$ ...
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1answer
17 views

projection into the intersection of two sets

Is projecting into a set $X$ and then projecting the result into the second set $Y$ the same as projecting into the intersection of $X$ and $Y$? I am not sure if these information will be needed to ...
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On Hilbert space, for projections $P,Q$, $P+Q$ is a projection if and only if $\text{ran}P\perp \text{ran}Q$

I am going through Functional analysis text by J.Conway, and encountered with next problem (2.3.4) : Let $P$ and $Q$ be projections. Show $P+Q$ is a projection if and only if $\text{ran}P\perp \...
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1answer
20 views

Solving an unrotated camera given known world and image points in the floor plane

Iʼm trying to find the intrinsic and some of the extrinsic parameters of an ideal pinhole camera, based on four chosen points for which I know both world and image co‑ordinates. I realize camera ...
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1answer
46 views

Equations of the orthogonal projection on the line $r = y - 2x + 1 = 0$

I'm trying to calculate in the Euclidean affine plane with respect to an orthonormal reference the equations of the orthogonal projection on the line r of equation y - 2x + 1 = 0. So, let $\phantom{3}...
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1answer
19 views

If original set of vectors have zero mean, will the orthogonally projections of the vectors onto another vector have zero mean?

Consider vectors $x_1, \cdots, x_n \in \mathbb{R}^m$. Define the vector $\mu \in \mathbb{R}^m$ to be the mean of the vectors: $$ \mu = \frac{1}{n}\sum_{i=1}^n x_i $$ Assume that $\mu = 0$, the zero ...
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Question about hyperplane based hash functions for vectors

I have come across a nice paper, which presents hash functions (section 3), which map some input vector from $\mathbb{R}^d$ to a bit. The property of these functions is that that the probability of ...
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2answers
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Prove that $(\mu_1 \otimes \mu_2)\circ {\Pi_1}^{-1}=\mu_1$

Suppose $\Pi_1 :(\Omega_1 \times \Omega_2, \mathcal{F_1} \otimes \mathcal{F_2} ,\mu_1 \otimes \mu_2) \rightarrow (\Omega_1, \mathcal{F_1},(\mu_1 \otimes \mu_2) \circ {\Pi_1}^{-1})$ is a projection map ...
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21 views

Minimizing the distance to a subspace (orthogonal projection) if a norm is not induced by an inner product

Let $V = \mathbb{R}^n$ with the inner product $\langle\cdot,\cdot\rangle$ and $U \subset V$ a vector subspace. Then for $v\in V$ and $x \in U$ the inequality$$ ||v-x|| \leq ||v-u||$$ is satisfied for ...
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1answer
12 views

Range of a unitary transformed orthogonal projection

Let $X$ be a Hilbert space, $P$ an orthogonal projection in $X$ and $Q \in L(X)$ (i.e. $Q \colon X \to X$ is linear and continuous) a unitary linear transformation, i.e. $Q^*Q=Id_X= QQ^*$ ($Q^*$ ...
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1answer
86 views

Conic equation of Fisheye line projection

When using an equidistant fisheye projection of 3d world points to a 2d image plane, straight 3d lines appear as conic sections on the image. Given a 3d line defined by a point $P=(Px,Py,Pz)$ and a ...
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0answers
18 views

Project onto vector that is not starting at the origin

In the image below we can project vector $d$ onto $c$ using formula $$proj_c(d) = c \frac{c•d}{d•d}$$ But what if I need to project onto a vector that is not at the origin? How do I project vector $d$...
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1answer
20 views

projection of a vector onto a vector space

supppose that we have a vector space $$A=\left(\left[ \begin{array}{} x\\ y\\ w\\ z\\ \end{array} \right] : x-y+w=0 \right)$$ and we wanted to find the closest point to a vector $$x= \left[\...
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0answers
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How do I get the projection function for the gradient method?

I need to calculate the projection function for $$ \text{minimize}\quad g(v)=(1/2)v^{T}DD^{T}v-y^{T}D^{T}v$$ $$\text{subject to} \quad -\lambda\leq v \leq\lambda$$ but I don't know how. Could you ...
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1answer
36 views

How to find Skew Projection Operator onto Plane parallel to some vector?

I was trying to solve previous year question paper of competitive exam In that I observed some strange question which I have not encountered before. They had given one equation of plane and told to ...
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1answer
24 views

For every projection $p$ and normal $a$ in a C*-algebra $A$ (with $ap=pa$), there is a $*$-isomorphism $C(\sigma(a))\to C^{*}(a,p)$ such that …

Let $a$ be a normal element of a (non-unital) C*-algebra $A$. I am trying to prove that for every projection $p\in A$ that commutes with $a$ (i.e. $p=p^{2}=p^{*}$ and $ap=pa$), there is a $*$-...
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2answers
34 views

Given what a reflection matrix does on a subspace, find the subspace - Can't solve

I can't really solve this exercise I've been trying to solve for some time now. It goes like that: Matrix $R$ ($\in \mathbb R^{3\times3}$) is a reflection matrix, in relation to subspace $U$, $U=\...
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1answer
36 views

Project vector onto a discrete subspace

I apologise that my explanations aren't very rigorous; I hope you will still get the idea of the question. Let $\vec v =(v_1, v_2, v_3)$ be the vector to project. Let P be a plane defined by the ...
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1answer
30 views

The classical projection theorem

I am going over a proof of the classical projection theorem which states the following: Let $H$ be a Hilbert space and $M$ a closed subspace of $H$. Corresponding to any vector $x \in H$, there is a ...
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1answer
71 views

Non-monotone sequence of orthogonal projections

Let $\mathcal{H}$ be a separable Hilbert space and $A_m:\mathcal{H}\to \mathbb{R}^m$ linear with $\sup_m \|A_m\|<\infty$. Assume that for all $x\in\mathcal{H}\setminus \{0\}$ there is a $\...
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1answer
23 views

Best approximation of a function among closed linear manifolds

Let $H$ be an infinite-dimensional Hilbert space and consider a $n-dimensional$ closed linear manifold generated by a subset of orthonormal basis, say, $M = span(\{u_1,u_2,\cdots,u_n\})$. Of course, ...
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1answer
34 views

Ortogonal basis for a subspace of $R^{4}$ and find a point in a plane closest to the origin.

Let $V \subset R^{4}$ a subspace defined for the equation $x_{1}+3x_{2}-5x_{3}-x_{4}=0$. a) Find a ortogonal basis for $V$. b) Wich point in the plane $x_{1}+3x_{2}-5x_{3}-x_{4}=36$ is closest to $(...

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