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Questions tagged [orthogonality]

This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.

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Finding orthogonal basis in $\mathbb R^4$ from given vectors

I have two subsets of $\mathbb R^4$ $S=((-1,0,1,1),(0,1,1,1),(1,0,0,1))$ and $T=(x,y,z,x-y+2z)$ I've proved that T is a subspace of $\mathbb R^4$ and that S is a basis for T. So far, so good! I ...
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2answers
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The Order of Orthogonality [on hold]

I would like to show that $B\subset A$ implies $A^{\bot}\subset B^{\bot}$. Note the meaning behind this: The bigger a subset, the smaller its orthogonal should be. Let $x$ be in the complement of A. ...
2
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1answer
43 views

Finding orthonormal basis. Is there error on textbook?

The problem is finding orthonormal basis for W=span{u1=x,u2=x^2} And as lots of people think, it is not very difficult problem My answer is ${ \sqrt{3}x,\sqrt{80}(x^2-\frac{3}{4}x) }$ But, ...
3
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1answer
41 views

$H$ Hilbert space, $T$ symmetric bounded linear, when is $H=R(T) \oplus N(T)$?

I just saw in an exercise that if I have a prehilbert space $H$ and $T$ a linear, bound and symmetric operator then $R(T)=N(T)^{\perp}$. Now I was asking myself whether $H=R(T) \oplus N(T)$. On wiki I ...
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0answers
13 views

orthogonal complement of subspace

I am not a mathematician but I faced the following problem when I want to apply some machine learning algorithm to solve nlp problem. let say that I have a matrix $A$ that is $100X50$ and I project ...
0
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1answer
41 views

Find the equation of the plane that passes through the points

Find the equation of the plane that passes through the points: $$P_1=(1,1,2)\\P_2=(2,3,3) \\P_3=(3,-3,3)$$ The answer writes: Let $x=\vec{P_1P_2}=\begin{bmatrix}1\\2\\1\end{bmatrix}$ and $y=\vec{...
1
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1answer
20 views

Find orthogonal projection of $ [n,0,0,…,0]^T$ on subspace $V$

$n>1$ Given is $$V = \left\{ \vec{x} \in \mathbb R^n : x_1+x_2 + ... + x_n = 0 \right\} $$ a) Find orthogonal basis of $V^{\perp} $ b) Find orthogonal projection $\vec{x} = [n,0,0,...,0]^T$ on ...
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2answers
40 views

Find the orthogonal bases of the space $ V $ and $ V^{\perp}$

In space $\mathbb R^3 $ f Find the orthogonal bases of the space $ V $ and $ V^{\perp}$ where $$V = \left\{ \vec{x} \in \mathbb R^3 : x_1 - 3x_2 + x_3 = 0 \right\} $$ On the begining, I know that may ...
1
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1answer
36 views

Orthogonal complement of certain family of functions.

We know that the linear span of subspace $A$ in Hilbert space $X$ is dense if orthogonal complement of $A$, that is $A^{\bot}$ is trivial. I am given a family of functions $A=\{f_n\}$ and I was trying ...
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0answers
49 views

Prove that there exists a semi-orthogonal $U$ such that $U^TAU=B$, where $A$ and $B$ are positive-definite symmetric matrices.

Let there be a semi-orthogonal matrix $U \in \mathbb{R}^{m\times n}$ such that $U^TU=I_n$ if $m > n$ If $A \in \mathbb{R}^{m\times m}$ and $B \in \mathbb{R}^{n\times n}$ are positive-definite ...
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0answers
55 views

Projection associated to the decomposition $H=M⊕N$

Let $H$ be a Hilbert space and let $M$ and $N$ be two closed subspaces in $H$ such that $H=M⊕N$. I'm trying to find a formula giving $P_{M,N}$ (the projection onto $M$ with respect to $N$) in terms ...
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0answers
13 views

Total subsets in incomplete inner product spaces [duplicate]

I'm reading Induced Representations of locally compact groups by Kaniuth and Taylor and I don't understand how total subsets work (in particular in Lemma 2.24). I know that a total subset of an inner ...
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0answers
19 views

The definition of orthogonal complement in the column space

Denote $A^\bot$ is the matrix satisfied $A'A^\bot=0$ with the highest rank. Proof that: (1)$I-(A')^-A'$ is a $A^\bot$, here $A^-$ means pseudo inverse. (2)$M(A^\bot)=M(A)^\bot$. $M(A)$ is the column ...
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0answers
31 views

How to determine functions orthogonal to $\sin(a (T-t))$?

How to determine functions $u \in L^2 \cap L^\infty[0,T]$, $T > 0$, that are orthogonal on $[0, T]$ to $\sin\left[\frac{\pi^2 n^2}{l^2}\alpha \left(T - t\right)\right]$? In other words, how to ...
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2answers
20 views

Basis for orthogonal complement possibly with linear combinations

In $\mathbb{R}^4$, consider the subspace $W = Span(u_1, u_2,u_3)$ with $$u_1 = (-1,1,0,0)$$ $$u_2 = (-1,0,1,0)$$ $$u_3 = (-1,0,0,1)$$ $(a)$ Use the Gram-Schmidt Process to ...
2
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1answer
36 views

Inner product with orthogonal complement

Let $\mathbb{R^3}$ be equipped with the inner product $<,>$ defined by setting $$<\mathbf{u},\mathbf{v}>=2u_1v_1+5u_2v_2+3u_3v_3$$ for any pair of vectors $\mathbf{u}=(u_1,u_2,u_3)$ ...
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4answers
34 views

Orthogonal complement and norm

Can someone point out what I am fundamentally doing wrong in this question? Consider the vector space $\mathbb{R}^3$ with the standard inner product (dot product) and let $H=span\left\{(2,1,0),(0,1,...
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1answer
31 views

On the construction of orthogonal polynomials

In the following proof, argument goes on based on considering $C_n$ to be nonzero then it finishes the proof for $C_n=0$ : Also if we set $C_n=0$ in Eq. (6.10) then must $m=0,1,2,...,n- 2$ in Eq. (6....
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1answer
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Why is the dot product of two columns $j$ and $k$ in FFT equal to $1+w^{j-k} + \dots + w^{(n-1)(j-k)}$?

The matrix is of the following form: $\begin{pmatrix} 1 & 1 & \cdots & 1 & 1 \\ 1 & w & \cdots & w ^{n-2} & w^{n-1} \\ \vdots & \vdots & \ddots & \vdots &...
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1answer
35 views

Existence of orthogonal matrices with zero diagonal and non-zero off-diagonal values

Does there exist an orthogonal matrix whose diagonal values are all zero but whose off-diagonal values are all non-zero for any $\Bbb R^n$? Furthermore, does this conclusion change if we are talking ...
1
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2answers
56 views

Why $\int_0^{2\pi} \sin^2 x \mathrm{d}x \neq \int_0^{2\pi}\sin x \sin nx \mathrm{d}x \to n=1$

By orthogonality, we know that $$\int_0^{2\pi}\sin mx \sin nx \mathrm{d}x = \pi$$ iff $m=n$ and $0$ otherwise. Nevertheless, when I am calculating it as follows, I get $0$. $$\int_0^{2\pi} \sin x \...
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1answer
27 views

Infinite linear combination

I don't know how to start the proof of the following statement: Let $V$ a real vector space with inner product $\langle\cdot,\cdot\rangle$. Consider a subset $C=\{v_j\}_J\subset V$ such that $\...
2
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2answers
27 views

$v \subseteq H \implies V^\bot$ is a closed subspace of the Hilbert space $H$

Exercise : Show that if $H$ is a Hilbert space and $V \subseteq H$, then $V^\bot$ is a closed subspace of $H$. Attempt : I thought of two possible approaches. One would be a classic one, getting ...
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0answers
15 views

Orthogonality of integer shifts and sum of fourier transforms

A function $\psi \in L_2(\mathbb{R})$ is orthogonal to all integer shifts of a function $\varphi \in L_2(\mathbb{R})$ if and if only $$\sum_{k\in \mathbb{z}} \hat{\varphi}(\xi+k)\overline{\hat{\psi}(\...
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1answer
10 views

Orthogonal transformation of standard normal sample

I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf Shouldn't $Var(Y_i)=\sum_{k=1}^nv_{ki}^2$ (from how $Y_i$ is ...
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1answer
25 views

Linear Algebra: orthogonality in $\mathbb{R}^{3}$

$$v_{1} = (1, 1, 1)$$ $$v_{2} = (-1, 1, 0)$$ $$v_{3} = (-1, -1, 2)$$ The vectors $v1$, $v2$, and $v3$ form an orthogonal basis for $\mathbb{R}^{3}$. If $w = (4, -2, 4)$, then what is the ...
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1answer
28 views

Are circles also squares in $(\mathbb{R}^2,||\cdot||_{\infty})$?

In $(\mathbb{R}^2,||\cdot||_{\infty})$ circles appear to be squares. But do squares exist in general normed space when we do not have an inner product, hence no natural notion of orthogonality and ...
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0answers
33 views

Show that these curve families are orthogonal: $f(xy) = C$ and $y^2 - x^2 = D$ [duplicate]

Let the function $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable and its derivative is never zero. Show that these curves are orthogonal: $$f(xy) = C$$ $$ y^2 -x^2 = D$$ $C$ and $D$ are ...
2
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1answer
43 views

Intersection of n hyperplanes in $\mathbb{R}^n$

For all unit vector $\nu \in \mathbb{R}^n$ consider an affine hyperplane $A_{\nu}$ orthogonal to the direction $\nu $. Now consider n linearly independent unit vectors $\nu_ 1 , \nu_2, \dots, \nu_n \...
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1answer
22 views

Let $S$ be the subspace of $\mathbb{R}^4$ spanned by $x_1=(1,0,-2,1)^T$ and $x_2=(0,1,3,-2)^T$. Find a basis for $S_\perp$

Let $S$ be the subspace of $\mathbb{R}^4$ spanned by $x_1=(1,0,-2,1)^T$ and $x_2=(0,1,3,-2)^T$. Find a basis for $S_\perp$. For this kind of question, if the subspace is spanned by one vector, I know ...
0
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1answer
25 views

Let $S$ be the subspace of $\Bbb R^3$ spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$

(a) Let $S$ be the subspace of $\Bbb R^3$ Spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$, let $A =\begin{bmatrix}x_1 &x_2 & x_3 \\ y_1 &y_2 &...
1
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1answer
55 views

Show that $\langle x,y \rangle = 0$. [duplicate]

Suppose that $\left (X, \langle \cdot,\cdot \rangle \right)$ be a complex inner product space. Let $x,y \in X$ be such that $\|\alpha x + \beta y \|^2 = \|\alpha x\|^2 + \|\beta y\|^2$ for all pairs $\...
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3answers
48 views

Can we say that $x \perp y\ ?$ [closed]

Suppose $\left (X, \left < . ,. \right > \right)$ be a Hilbert space over $\Bbb C.$ Let $x,y \in X$ be such that $\|x+y\|^2 = \|x\|^2 + \|y\|^2$ $($ where $\|.\|$ is the norm induced by the ...
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0answers
16 views

Complex Endomorphism $f$ with orthogonal Eigenspaces implies that $f$ is unitary

I already posted this question earlier today without much success. After 3 hours and only 10 views I decided to give it another shot :) Of course I'll delete the second post! Let $V \ne 0$ be an ...
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2answers
39 views

Finding perpendicular unit vector in $\mathbb{R}^n$ to hyperplane

Suppose I have $n-1$ linearly independent vectors $(v_1, ..., v_{n-1})$ in $\mathbb{R}^n$ that together form a basis of a hyperplane. I'm looking to find a last vector $v$ that is normal to the ...
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1answer
40 views

Some subspaces are either closed or dense

a) Let $a_{n}\rightarrow 0$ and $M=\{x\in \ell^{2}:\sum a_{n}x_{n}=0\}$. Show that the subspace $M$ of $\ell^{2}$ is closed or dense according as $% \{a_{n}\}\in \ell^{2}$ or $\{a_{n}\}\notin \ell^{2}$...
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1answer
17 views

How to prove symmetric matrix is orthogonally diagonalizable?

I learned below theorem and there is a proof that orthogonally diagonalizable matrix is symmetric, but there is no proof that symmetric matrix are orthogonally diagonalizable. Theorem 2. An $n\...
2
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1answer
43 views

Damped vibrations of a membrane stretched over a circular frame

I am given this following PDE with the initial and boundary conditions with $0 < r < 1$, $t > 0$, and $v_0$ being a constant: $u,_t,_t + 2bu,_t = u,_r,_r + \frac{1}{r} u,_r$ $u(t,r=0) = 0, \...
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0answers
19 views

converting trig function to orthogonal coordinates

How can a trig function (e.g. $r=\cos\theta$) be converted to orthogonal coordinates. I'm imagining an infinite number of vector pairs forming $90^\circ$ angles but that doesn't feel right.
2
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1answer
60 views

Do every two orthogonal matrices in $\text{SO}(n)$ lie in the same coset of $\text{SO}(2)$?

Let $A,B \in \text{SO}(n)$. Does there exist a homomorphism of Lie groups $\phi:\text{SO}(2) \to \text{SO}(n)$, such that $A,B$ lie in the same coset of $\phi(\text{SO}(2))\le \text{SO}(n)$?
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1answer
16 views

Equality of orthogonal projection norms. [closed]

Let $F,G$ be two sub vector spaces of $\mathbb{R}^n$, and denote $P_F,P_G$ the orthogonal projection matrices on $F$ and $G$ respectively. Suppose that for all $v \in \mathbb{R}^n$ we have: \begin{...
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0answers
36 views

For which integers $k$ there exists an $(k-1)$-$(k, k, 1)$ orthogonal array?

This is what I've found out so far: Since the first $k-1$ columns must have all the possible $k^{k-1}$ strings of length $k-1$ in exactly one row and the rows can be reorganized without changing the "...
0
votes
1answer
20 views

How to show that the bipolar co-ordinates are othogonal

How to show that the bipolar co-ordinates are othogonal where $x=\dfrac{\sin hv}{\cos hv-\cos u},y=\dfrac{\sin u}{\cos hv-\cos u},z=z$ where $u\in [0,2\pi]$ and $y,z\in (-\infty,\infty)$. ...
1
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3answers
74 views

Determining a vector orthogonal to $q_1=(1,1,1)$ and $q_3=(1,1,-2)$, why I'm wrong with my calculations?

Consider the vectors $q_1=(1,1,1)$ and $q_3=(1,1,-2)$. I need to find a third vector $q_2$ such that $\{q_1,q_2,q_3\}$ is a arthogonal basis for $\mathbb{R}^3$. My problem is the following: I did ...
0
votes
1answer
31 views

Orthogonal complement not resulting ok

I have the subspaces: $$S = \langle(2,1,-1), (-1,2,0)\rangle, \qquad T = \{ X + Y + 2Z =0; X - Y - Z = 0\}.$$ I got that $T = \langle(1, 3,-2)\rangle$. All vectors are linearly independent, so $S +...
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0answers
13 views

$L^{-1}$ in $LU$ decomposition relationship with $Q$ in $QR$ decomposition

Playing around with matlab, I see that the last row of $L^{-1}$ where $L$ is the part of the $LU$ decomposition is colinear with the last column of $Q$, in the matrix's $QR$ decomposition. Is there a ...
0
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0answers
19 views

Orthogonal complement of $H_a =\left\{g \in V: g\left(t+\frac{1}{\sqrt{2}}\right)=g(t) \right\}$

If $\;\;V=\{ f:\mathbb{R}\rightarrow \mathbb{C} |\; f \text{ is continuous and has period }1\}$, $\;\; \langle f | g \rangle$ is defined as $ \displaystyle \langle f | g \rangle = \int_0^1 \overline{f(...
1
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1answer
68 views

Orthogonal complement of $H_a =\left\{g \in V: g\left(t+\frac{1}{2}\right)=g(t) \right\}$

If $\;\;V=\{ f:\mathbb{R}\rightarrow \mathbb{C} |\; f \text{ is continuous and has period }1\}$, $\;\; \langle f | g \rangle$ is defined as $ \langle f | g \rangle = \int_0^1 \overline{f(t)}g(t)dt$, $\...
0
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0answers
29 views

Consider the plane P in R-3 given by x-y-2z=0

I found the matrix A whose columns are a basis for P, A=[1,-1,-2] (vertical form). Using that I was able to find the projection matrix: P=$\frac{-1}{2} \left( \begin{array}{cc} 1 & -1 & 2 \\...
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0answers
23 views

Orthogonality of generalized Newton symbol

Consider the functions $P_{n}(x)={x \choose n}.$ My question is, if there exists a measure $\mu$ with support being a subset of $(0,\infty)$ such that the family $\{P_{n}\}$ is orthogonal in $L^{2}(\...