For questions related to orthonormality. A set of vectors in an inner product space is called orthonormal if each vector is a unit vector, and vectors are pairwise orthogonal.

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60 views

Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting

I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
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0answers
26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...
0
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0answers
18 views

Orthonormal Basis and evaluation of expression

Let the vectors $\mathbf{a}$ and $\mathbf{b}$ be specified by $$\mathbf{a} = -3\mathbf{e_1} + 2\mathbf{e_2} - 8\mathbf{e_3}$$ $$\mathbf{b} = 5\mathbf{e_1} - 7\mathbf{e_2} + 3\mathbf{e_3}$$ where ...
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0answers
24 views

Normalizing disturbed rotation matrix

I was playing with simulations of Euler's equations of rotation in this question. This involves integrating an ordinary differential equation of a rotation matrix, $R$, which is calculated for all of ...
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0answers
15 views

Coefficients of Fourier-Bessel series for a Neumann condition

What is the expression for coefficients of Fourier-Bessel series for a Neumann condition? I know what it is for Dirichlet condition. $\frac{\partial f}{\partial x} = 0$
2
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1answer
26 views

Self Adjoint Linear Maps and Matrices with respect to orthonormal bases

Let $T: V \rightarrow V$ be a self adjoint linear map where $V$ is an inner product space. Known fact: With respect to any orthonormal basis- the the matrix for $T$ is conjugate symmetric. When ...
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1answer
9 views

Proof of Hesse Normal Form

Let $F$ be a hyperplane in $\mathbb{R}^n$. Let $0 \neq a = \begin{pmatrix} x_1 \dots x_n \end{pmatrix}^\top$ and $F=\{x\in \mathbb{R}^n \mid a \bullet x=0\}$. If we use $N=\frac{a}{\mid a \mid}$ we ...
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1answer
23 views

Orthonormal series of complex functions

Let $$\chi_n(z) = \frac{1}{\sqrt{2\pi}}z^{n-1}.$$ Prove $(\chi_n)$ is orthonormal on $\partial B(0,1)$ in regard to $$\langle f(z), g(z)\rangle = \int_{\partial ...
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1answer
21 views

Eigenfunctions for the symmetric kernel of an integral equation

The solution of the symmetric integral equation below: $$g(s) = f(s) + \lambda \int_{-1}^{1} (st +s^2t^2)g(t)dt \tag{$*$}$$ with separable kernels method is $$g(s) = f(s) + \lambda \int_{-1}^{1} ...
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2answers
42 views

Eigenvalue lower bound for this simple matrix

For orthonormal vectors ${\textbf q}_1, {\textbf q}_2, {\textbf q}_3 \in \mathbb{R}^3$, I want to show that the matrix $$\big(\begin{array}{c:c:c}2{\textbf q}_1 & -{\textbf q}_2 & -{\textbf ...
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3answers
63 views

Find orthonormal basis

I'm trying to solve the following exercise in my book: Find an orthonormal basis $\alpha$ for the vector space $\left(\mathbb{R},\mathbb{R}^{2 \times 2},+\right)$ (with default inner product, ...
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0answers
29 views

What is a biorthogonal system?

What does biorthogonal mean ? If they say let the system $l^1,l^2,l^3,...,l^n$ Be biorthogonal to the bases $x_1,x_2,x_3,...,x_n$ Of the kernel of $Λ$ So that $Λ$ Is a linear operator
2
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2answers
35 views

Vector analysis: understanding formulas for normal and tangent

I'm studying a chapter on vector analysis and I don't really understand why the following equations are switched for the two situations. (My course is not in English, so I apologize if I translated ...
0
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1answer
23 views

How do I perform Gram-Schmidt on floating point vectors with epsilons in them?

Let $\epsilon$ be a small positive number such that $1+\epsilon$ and $3+2\epsilon$ are machine numbers but $3+2\epsilon + \epsilon^{2}$ is computed to be $3 + 2\epsilon $. Now, let the (classical) ...
2
votes
3answers
32 views

Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?

I am currently working on showing that $\Vert Px \Vert_{2} \leq \Vert x \Vert_{2}$, where $x \in$ an inner product space $X$, and $P$ is the orthogonal projection operator. Also, I am supposing that ...
3
votes
4answers
381 views

Prove that the orthogonal projection operator is idempotent

Let $\{u_{1},u_{2},\cdots,u_{n}\}$ e an orthonormal basis for a subspace $U$ in an inner product space $X$. Define the orthogonal projection of $X$ onto $U$, $P:X \to U$, to be $Px = ...
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1answer
37 views

Prove $||Px|| = ||x||$ for Orthogonal Matrices

If $P$ is an $n\times n$ orthogonal matrix, then prove that $\Vert Px\Vert = \Vert x\Vert$ I tried manipulating the expression arbitrarily, but I can never understand why should I be doing ...
0
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1answer
32 views

Why Must A Matrix be Symmetric for Orthogonal Diagonalization

So far, all we are doing in class is determine if the matrix A is symmetric, find the basis for the eigenspace P, and apply Gram Schmidt for it to be orthogonal. My question is; why must A be ...
1
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1answer
47 views

$\mathbb{C^2}$ treated as a real vector space

As a real vector space the dimension is 4. What is an orthonormal basis for it with respect to either the standard real inner product?. I've tried gram schmidt with the obvious choice of 4 vectors( ...
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0answers
29 views

Inner products over fields other than $\mathbb{R}$ or $\mathbb{C}$

Do inner products over other fields even make sense? E.g mod 7 or something. Also if I have a complex vector space $V$ it is a also a vector space over the reals but has twice the dimension e.g ...
2
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2answers
32 views

Lemma: $x \cdot (Ay) = (Ax) \cdot y$

As simple as this may sound, I just do not understand what this statement implies. An $n \times n$ matrix A is symmetric if and only if: $$\bar{x}.(A\bar{y}) = (A\bar{x}).\bar{y}$$ Why is this ...
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0answers
24 views

Why does the Lanczos algorithm make an orthonormal basis for the Krylov subspace?

Starting with a $v_0$= $b_0$=0 and a symmetric positive definite matrix A. Why does the following algorithm forms an orthonormal basis span{$v_1$,$v_2$,...,$v_n$} for $K_n$(A,$v_1$)? for k=1,...,n-1 ...
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3answers
98 views

Basic conceptual questions about orthogonality in Linear Algebra,

I have the Gram-Schmidt algorithm memorized, so that I can always compute an orthonormal basis, when I need it (on pen and paper, I don't studying mathematical / scientific computing ... yet). Could ...
2
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1answer
37 views

Calculate orthonormal basis using Gram-Schmidt

Our professor gave this exercise to help us review the topic we covered in class, but it seems my knowledge is not sufficient (or we didn't cover it in enough detail during class). Assume we are ...
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1answer
71 views

Finding the least-squares solution of Ax = b when the columns of A are orthonormal.

Find a formula for the least-squares solution of Ax = b when the columns of A are orthonormal.
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0answers
23 views

Construct an Orthogonal basis for the subspace $W\subseteq R^3$. given by the following basis:

Construct an Orthogonal basis for the subspace $W\subseteq R^3$. given by the following basis: $\{\begin{pmatrix} 3 \\ 0 \\ -1 \\ \end{pmatrix} \}$, $\{\begin{pmatrix} 8 \\ 5 \\ -6 \\ \end{pmatrix} ...
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1answer
35 views

Proof for $u · v = (u · w_1)(v · w_1) + … + (u · w_n)(v · w_n)$ - Parseval's identity

Suppose {w1, w2, ... wn} was an orthonormal basis for Rn and u and v were vectors in Rn. I'm trying to prove that u · v = (u · w1)(v ·w1) + ... + (u · wnn)(v · wn) I know that since {w1, w2, ... wn} ...
2
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1answer
45 views

Unique decomposition of a vector

Show that every vector $\vec{u}$$\in$$U$ can be uniquely decomposed into $\vec{u}$$=$$\vec{u}_{1}$$+$$\vec{u}_{2}$ where $\vec{u}_{1}\in{W}\subset{U}$ and $\vec{u}_{2}\in{W^⊥}$.
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1answer
43 views

Prove that a Vector Orthogonal to an Orthonormal Basis is the Zero Vector.

Stuck on this proof. Let W be an inner product space (with unspecified inner product, $<\vec x, \vec y>$), and with orthonormal basis B = {$\vec w_1, \vec w_2, \ldots ,\vec w_n$}. Suppose ...
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0answers
13 views

Unitary and non-unitary

I have a problem where the optimum is achieved when non-unitary is equal to the unitary? Given the unitary matrix $\mathbf{U} \in \mathcal{C}^{N \times d}, N>d$, $\mathbf{G} \in \mathcal{C}^{N ...
2
votes
3answers
62 views

Orthonormal Basis, Eigenvectors and Eigenvalues in $\mathbb{R}^n$

Let $n$ be a positive integer and let $c_1,..., c_n$ be a list of real numbers. Let $\{v_1, . . . , v_n\}$ be an orthonormal basis for $\mathbb{R}^n$, let $d = \text{min}\{c_1,...,c_n\}$. For each $1 ...
0
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1answer
35 views

Normalizing an Orthogonal Set of Vectors

I've been given the basis $B=\{(1,-1),(1,-2)\}$ and am asked to find the orthonormal basis using the Gram-Schmidt process. No problem! I have the orthogonal set of vectors $w_1 = (1,-1)$ and $w_2 = ...
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0answers
16 views

Get the orthonormal basis and the matrix of the orthogonal projection

Let $U$ be the subspace of $\mathbb R^3$ that concides with the plane through the origin that is perpendicular to the vector $n$ = (1, 1, 1) $\in \mathbb R^3$. (a)Find a orthonormal basis for $U$. ...
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2answers
41 views

How do I show that W is an orthogonal basis?

Given that subspace $W = \text{span}\{[1 1 0 -1], [1 0 1 1], [0 -1 1 -1]\}$, how do I show that $W$ (not subspace $W$) is an orthogonal basis? Any help would be greatly appreciated.
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0answers
41 views

How to diagonalize the matrix of a linear map $F$ without calculating $F$?

I know how to do 4a) and found the orthonormal basis.. I believe the only eigenvalues of $F$ are $-1$ and $1$ as they will result in either a reflection or the same result. However, I'm ...
0
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2answers
43 views

How to make an Eigenvector orthonormal?

I am trying to figure out the PCA of a Data-set using calculation , and in one phase of this calculation I have the two eigenvectors : $V_1=(\frac{1}{\sqrt2} , -\frac{1}{\sqrt2} ,0)t $ ; $V_2 = ...
4
votes
2answers
45 views

Is every hyperplane in $\mathbb{R}^n$ determined by a unique normal vector?

Is every hyperplane in $\mathbb{R}^n$ determined by a unique normal vector? And why? I analysed for $\mathbb{R}$, a hiperplane in $\mathbb{R}$ is a point, so the hyperplane is $PX= \alpha$, with ...
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2answers
34 views

How to normalize an eigenvector when it has $\sqrt{i}$ as an entry?

If $V=\mathbb{C}^2$ and $T(a,b)=(2a+ib,a+2b)$ I found $$[T]_\beta=\begin{bmatrix}2 & i \\ 1 & 2\end{bmatrix}$$ hence the eigenvalues are $2+\sqrt{i},2-\sqrt{i}$. So using this, eigenvectors ...
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0answers
18 views

Prove $\left \| f-L_{k}^{s}f \right \|_{2} = min_{q \epsilon V_{k} }\left \| f-q \right \|_{2} $

Let q be arbitrary and consider the quadratic function of t defined by: $\phi (t)=\left \| f-L_{n}^{s}f+tq \right \|_{2}^{2}$ Note: $L_{k}^{s}f = \sum_{i=1}^{k}(f,p^{i})p^{i}$ for $i = 1,...,k$ ...
0
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2answers
42 views

How to prove the following assertion about Hilbert spaces.

Let $X$ be a separable Hilbert space and $Y$ a closed subset. Then I want to conclude that $Y$ is separable. So I wanted to invoke the theorem that says that a Hilbert space is separable iff it has a ...
0
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1answer
61 views

Compute the angles between the elements of the standard basis with respect to this scalar product

I have an homework to solve but I am behind with the theory. It doesn't look difficult, since it is about computations, just it would be good for me to find a short document which explains exactly ...
0
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2answers
49 views

Proof involving the inequality $\sum_{i=1}^n \langle v_i, w\rangle^{2} = \|w\|^2$

Let $V$ be a finite dimensional vector space with an inner product $\langle . \rangle$. Let $v_1,v_2,...,v_n\in V$, $v_i\ne 0 \quad \forall i \quad$, so that $ \forall w\in V$ we have that: ...
0
votes
1answer
62 views

Showing $\sin(nx)$ is a complete orthonormal system

I want to prove that the system $\sin(nx)$ for $n=1,2,\cdots$ is complete in $L_2[0,\pi]$, so what I do is assume that: $$\int_0^\pi f(x)\sin(kx)dx=0,\quad k=1,2,\cdots$$ An define an odd function: ...
0
votes
0answers
50 views

Orthonormal basis - Gaussian integral

For the following Euclidean vector spaces $(V,\omega)$ and a basis $f_1,\ldots,f_n$, run Gram-Schmidt orthonormalisation process to arrive at an orthonormal basis. (i) $V=\mathbb R^3$ with ...
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0answers
28 views

From orthogonal group to orthonormal frames

Define the orthonormal frame of a space $R^n$ to be a set of vectors ($b_1$, $b_2$, ..., $b_n$) if these vectors together form an orthonormal basis of $R^n$, and denote this frame to be $F_n$. ...
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1answer
184 views

How to find an Orthonormal Basis for Null( A$^T$ )

I'm studying for an exam and I'm not sure how to do this. I understand what the definitions mean (for the most part) but I'm not sure how to apply it to the problem. Let A = ...
5
votes
1answer
62 views

Every complete orthonormal set in a Hilbert space $H$ is an orthonormal basis, if and only if $H$ is finite dimensional.

Show that any orthonormal set in a Hilbert space $H$ is linearly independent, and use this to show that $H$ is finite dimensional if and only if every complete orthonormal set is an orthonormal basis. ...
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1answer
48 views

Does a small perturbation of an orthonormal basis create strongly linearly independent vectors?

Let $e_1, e_2, \ldots$ be an orthonormal base in the separable Hilbert space $\mathcal{H}.$ Let $\psi_1^n, \psi_2^n, \ldots \in \mathcal{H}, n\in \mathbb{N}$ be vectors such that $\sup_{i} \| ...
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vote
2answers
33 views

Every orthonormal set in a Hilbert space is contained in some complete orthonormal set.

Let $H$ be a Hilbert space. Show that every orthonormal set in $H$ is contained in some complete orthonormal set. I'm unable to start from any direction. Do I use the Gram-Schmidt orthogonalization ...
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votes
1answer
44 views

If two complex vectors are orthonormal, does conjugating one of them preserve orthonormality?

Suppose $x_1$ and $x_2$ are two orthonormal vectors with complex elements. I wonder if $x_1$ and $\bar x_2$ still orthonormal to each other? Thanks for any help!