# 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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1answer
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### Can it be said that $E(A_1)E(A_2) = E(A_2) E(A_1) = 0$ for $A_1 \cap A_2 = \varnothing\$?

Let $(X,\mathcal A)$ be a measurable space. Let $\mathcal H$ be a Hilbert space and $E : \mathcal A \longrightarrow \mathscr P (\mathcal H)$ be a projection valued map. For all $x \in \mathcal H$ with ...
2answers
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### Project a vector onto subspace spanned by columns of a matrix

From this question we know that if $x\in\mathbb{R}^{n\times1}$ is a vector, then the (normalized) outer product matrix $$\frac{x x^\top}{||x||^2}\, \in \mathbb{R}^{n\times n}$$ can operate on ...
0answers
32 views

### Regarding an isometric drawing of a circle, how does this align perfectly?

In the above image, the circle is drawn as an isometric view. Why does it align well when we draw a six-pointed star? The radius of the larger arc is thrice the radius of the smaller arc. My question ...
0answers
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1answer
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### Norms of orthogonal subspaces

Let V be a finite dimensional real inner product space and U, W subspaces of V such that U is orthogonal to W. Show that for any $v ∈ V$ $$||v||^2 ≥ ||proj U (v)||^2 + ||projW (v)||^2$$ Hello guys. I ...
1answer
37 views

### Prove that $A: (v_1)^\perp \to (v_1)^\perp ,\ e_j\mapsto Pv_j$'s determinant stays the same for all ONBs $e_j$, where $P$ is an orthogonal projection.

I am sorry for the scuffed title, but there are not enough characters. Here is the problem in full: Take a basis $(v_1,\dots,v_n)$ of $\mathbb{R}^n$ and $W=(v_1)^\perp$. Let $P$ be the orthogonal ...
0answers
23 views

### Projection onto a Line: why must the projection be a multiple of the vector $a$?

In the book: Linear Algebra and its application -Gilbert Strang-, when the projection $p$ of vector $b$ onto vector $a$ is explained, this is said: We want to find the projection point $p$. This point ...
0answers
40 views

### How to extract a component of integer vector / music interval? [duplicate]

(I will ask this question in musical terms, but this seems to be related to projecting integer vectors onto each other, which I'm unfamiliar with. Perhaps I'm just looking for some existing notation ...
1answer
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1answer
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### Are geodesic preserved by projections in hyperbolic space?

In a the euclidean space $\mathbb{E}^n$ a line $\alpha$ is mapped to a line $\pi(\alpha)$ by an othogonal projection $\pi:\mathbb{E}^n\to P$ to some plane $P$. Projections in the hyperbolic space ...
0answers
52 views

### Orthogonal projection into a sparse subspace with $s$ dimension

Traditional orthogonal projection of a given point $y \in \mathbb{R}^n$ into a closed and convex set $D\in \mathbb{R}^n$ is defined as the follwing: $$P_D(y)=\arg\min_{x \in D}||x-y||_2^2$$ Now ...
0answers
20 views

### Equations of 3D Projections

Is anyone familiar with these equations? Please, let me know.
1answer
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1answer
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### Hyperplane Reflection

Given the reflection $R_H(−2, 2, 2, −3) = (−4, 0, −2, 1)$, what is the reflection of $R_H(−1, −1, −1, 3)$, and how I can find it ? I tried to use $R_H = (2P_H b -I)b$, but I have no idea how to keep ...
2answers
29 views

### Is the globe distorted? [closed]

Since there is no any types of maps that do not distorted. The way we make the globe is to put the paper on a sphere. So, it should also be distorted. Am I right? and also If this is true then what ...
0answers
30 views

### Prove $\forall x_iR=\overline{\exists x_i\overline{R}}$

My working: Let $\forall x_iR$ and $\overline{\exists x_i\overline{R}}$ where k > 1 and 1 ≤ i ≤ k and R is relation on S. To prove $\forall x_iR=\overline{\exists x_i\overline{R}}$, we need to show ...
0answers
10 views

### Define an inner product so that the orthogonal projection is a given vector

Let $x_2\in\Bbb{R}^4$ and $x_1\in\Bbb{R}^2$ be fixed vectors. I am assuming $\Bbb{R}^4=\Bbb{R}^2\oplus\Bbb{R}^2$ and $x_1$ lies in the first factor. I need to find a necessary and sufficient condition ...
1answer
36 views

### Sequences of orthogonal projection in bounded Hilbert space

Let {$P_i$} $\subset$ B(H) (bounded Hilbert space) be the projections. Suppose that: If $P_1<P_2<...$ and there exists a projection P $\subset$ B(H)such that $\lim_{i \to \infty}P_i\zeta=P\zeta$...
0answers
72 views

### Projection operators onto convex subset decomposition

Suppose $H$ is a Hilbert space and there are closed convex sets $C,D \subset H$ such that every $h \in H$ can be written as $h= c + d$ where $c \in C$ and $d \in D$. Denote by $P_C$ the orthogonal ...
1answer
20 views

### Projecting onto the space of Upper-Triangular-ish Matrices

I want to solve the independent component analysis (ICA) problem for a single time domain channel of input. The problem is, in my formulation, the mixing matrix has a special structure. I want to know ...
0answers
134 views

### Calculate 3D Rectangle from 4 projected points on screen

Given 4 known projected points on the screen, I need to calculate a 3D rectangle where the 4 projected points coincide with the rectangle corners from the o ...
0answers
33 views

### Prove that $P = P_U$ if and only if $P$ is self-adjoint

I was reading Axler's Linear Algebra Done Right and the following question appears as exercise $11$ in chapter $7$, section A: Suppose $P \in \mathcal{L}(V)$ is such that $P^2 = P$. Prove that there ...
1answer
12 views

### Does an Orthogonal Projection map a basis of the space to a spanning set of the its subspace for a Hilbert Space? [closed]

For finite dimensional vector spaces it is quite easy to prove that a surjective linear operator $V\mapsto W$ maps a basis of $V$ to a spanning set of $W$. Is this property still true for linear ...
0answers
21 views

### Alternating Projection Algorithm for two sets with empty intersection

Given two closed convex sets $A,B\subset\mathbb{R}^n$, with $A\cap B=\varnothing$. Consider the alternating projection algorithm with arbitrary initial point $x_0$ and update rule: $y_k=P_A(x_{k-1})$ ...
1answer
48 views

### Sequence of Nested Projections in an arbitrary Normed Linear Space Converges to the Identity

I have seen similar questions on this site, most notably this one: Convergence of projections onto a nested sequence of subspaces of a Hilbert space, but they all include the Hilbert space assumption. ...
1answer
43 views

### If $Q \subseteq \mathbb{R}^{n-1}$ is the projection of the set $P \subseteq \mathbb{R}^n$, are $Q$ and $P$ the same set except for 1 dimension?

Let $P$ denote a polyhderal set of $x$ values in $\mathbb{R}^n: \{x: Ax \leq b\}$. Let $Q$ denote the projection of $P$ onto $\mathbb{R}^{n-1}$ (i.e., $Q \subseteq \mathbb{R}^{n-1}$). Do the sets $P$ ...
0answers
35 views

### Orthogonal decomposition of a vector

I have the following question here. Let $W$ be the subspace of $\mathbb{R}^5$ spanned by the vectors $$\{(1,2,2,4,-1),(1,2,-1,1,1),(0,0,-1,-1,1),(0,0,1,1,0) \}$$. a) Find an orthogonal basis for $W$. ...
0answers
30 views

### An approach to handle an orthogonal projection onto a nonconvex set

I have a composite problem in hand, which can be expressed as \begin{align} \min_{x \in \mathbb{R}^n} \ \ h(x) + I_{C}(x), \end{align} where $h(x)$ is $L$-smooth and $I_{C}(x)$ is an indicator ...
1answer
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### Show that $\|\pi_c(y)-x\|_2^2 + \|\pi_c(y)-y\|_2^2 \leq \|x-y\|^2$, $\forall x$ in closed convex set $C$

Let $\mathcal{C}$ be the projection operator onto a closed convex set $\mathcal{C}$. Prove $\|\pi_c(y)-x\|_2^2 + \|\pi_c(y)-y\|_2^2 \leq \|x-y\|^2$, $\forall x \in \mathcal{C}$ In which $\pi_c(y)$ is ...
1answer
38 views

### Showing existence of a projective [duplicate]

Let $X$ be a normed space and $Y$ be a finite dimensional subspace of $X$. Show that there is a projective $P\in B(X)$ such that $Im P=Y$. Hint: First Solve for $dimY=1$ then generalize the solution ...
1answer
20 views

### For a family of projections how to prove $\vee(I-E_a) \ge I-\wedge E_a$?

In the book "Fundamentals of the Theory of Operator Algebras" by Richard V. Kadison and John R. Ringrose at page 111 (a screenshot is attached below) the authors claims: Since the map \$E \...