# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...