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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Normalizing a basis

Let the basis $B = \{1,x,x^2\}$ which is orthogonal. Now, I've seen the following: $$\|1\| = \sqrt {\langle 1,1\rangle} = \sqrt {4\cdot 1\cdot 1} = 2 $$ $$\|x\| = \sqrt {\langle x,x\rangle} = ...
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$\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2$

I want to show that $\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2((0,1); \mathbb{C}) =: X$. Of course the only problem is to show completeness. In our lecture we ...
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1answer
54 views

Proving that an orthonormal system close to a basis is also a basis

Let $\mathcal{H}$ be a Hilbert space and $(e_n)_{n \in \mathbb{N}} \subseteq\mathcal{H}$ be an orthonormal basis and $f_n$ be an orthonormal system such that $(f_n)_{n \in \mathbb{N}} ...
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1answer
12 views

Equivalent definitions of an orthonormal function

I want to prove that the following two definitions for an orthonormal function $\phi$, in terms of $kT$ time shifts, are equivalent. So let $T$ the symbol period and $k$ an integer. Definition 1 ...
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Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...
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1answer
51 views

Orthonormal Basis of a function

An Orthonormal Family $\{e_k\}_{k\in\mathbb{N}}$ is a basis if and only if $$f=\sum^\infty_{n=1}\hat{f}(n)e_n \ \ \ \text{in} \ \mathcal{L}^2(\mathbb{R})$$ where $f\in\mathcal{L}^2(\mathbb{R})$ ...
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Why an orthonormal polynomial set over a continuous domain is not over a discrete one?

I would like to read the proof showing that a orthonormal polynomial set over a continuous domain is neither orthonormal nor complete over a discrete values on that domain. For example, Zernike ...
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77 views

Why are there infinitely many orthonormal vectors?

By Graham Schmidt process we can create infinitely many orthonormal vectors, but my doubt is that why is it not bounded by the dimensionality of the space ? Intuitively (geometrically) how can we ...
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1answer
23 views

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$.

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$. Hint: Take a Cauchy sequences $(x_r)$, where $x_r=\sum ...
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30 views

Expansion coefficients of an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ for $\sum|c_n|^2$ to converge.

Expansion coefficients with respect to an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ in order that $\sum|c_n|^2$ may converge. Is this true or false? Give a proof or ...
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1answer
26 views

A unitary space, interpreted as a Euclidean space

Let $(V, \gamma)$ be a $n$-dimensional unitary space. Let $V_{\mathbb{R}}$ be the vector space $V$, interpreted as a $2n$-dimensional $\mathbb{R}$-vector space. I first want to show that ...
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1answer
19 views

Using the definition of unitary / orthogonal operators explicity for matrices:

If A is unitary, then $$AA^* = A^*A = I, and\ A^* = A^{-1}$$ I want to see this explicitly for a very simple unitary matrix, say, take the column vector A = (1,0,0) and we regard this as a 3x1 ...
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1answer
40 views

Help on understanding this process of getting orthonormal basis

This method was taught by someone and I am trying to understand the process. Assume that we have $(1,2,3)$ as first vector. Since we need another vector that will make the dot product $= 0$ with the ...
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0answers
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Bipolar toroidal coordinates - position vector, velocity and acceleration

Bipolar toroidal coordinates: $x = a \frac{\sinh\tau \cos\phi}{\cosh\tau-\cos\sigma}$ $y = a \frac{\sinh\tau \sin\phi}{\cosh\tau-\cos\sigma}$ $z=a \frac{\sin\sigma}{\cosh\tau-\cos\sigma}$ Would ...
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1answer
21 views

A linear operator T is normal if and only if there $[T]_\beta$ is normal, where $\beta$ is an orthonormal basis.

I am studying for a final exam and came across a sentence in my linear algebra textbook stating that "a linear operator T is normal if and only if there $[T]_\beta$ is normal, where $\beta$ is an ...
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1answer
29 views

Linear map, inner products and orthonormal bases

If $L: \mathbb{V} \to \mathbb{V}$ is any map and we have two inner products defined on $\mathbb{V}$, $[ , ]$ and $⟨ , ⟩$, and we pick two orthonormal bases w.r.t. each of these inner products, how can ...
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19 views

Linear map and inner products

How can one check that a map $L: \mathbb{V} \to \mathbb{V}$ preserves an inner product, provided that the basis with respect to the inner product is orthonormal?
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Orthonormal with respect to an inner product

What does it mean for a basis to be orthonormal with respect to an inner product?
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93 views

How many orthogonal matrices are there

this might sound like a stupid question, but what I mean is: You need $n \times n$ elements to define a square matrix $\in R^{n \times n}$. How many element do I need to define an orthogonal matrix? I ...
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4answers
47 views

Prove that $\|v \|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$

Suppose $(e_1,\cdots, e_m)$ is an orthonormal basis in $V$. Let $v \in V$ . Prove that $\|v\|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$ Let $v\in V$ and ...
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3answers
20 views

Completing an orthonormal basis of a plane to a basis for $\mathbb{R}^3$

I was asked to find an orthonormal basis for the plane $x + 2y +3z =0$. I found a regular basis, $(-2,1,0),(-3,0,1)$, and then performed the Gram-Schmidt process to find 2 orthonormal vectors that ...
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1answer
33 views

Is there anyway to check I have an an orthogonal and/or orthonormal basis?

I'm reading about Gram-Schmidt procedure in 3 dimensions. From what I understand the idea is to "fix" one of the vectors and alter the other 2 so they are all perpendicular. So say i have three ...
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1answer
31 views

Prob. 4, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $(e_n)$ be an orthonormal sequence in an inner product space $X$. Then, for every $x \in X$, we have $$ \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 \ \leq \ \Vert x \Vert^2.$$ Now ...
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1answer
41 views

Prob. 3, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to derive the Schwarz inequality?

Let $\left( e_n \right)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X$, we have $$ \sum_{n=1}^\infty \left\vert \langle x, e_n \rangle \right\vert^2 \ \leq \ ...
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27 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
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0answers
20 views

Maximally distant orthogonal matrices

I would like to construct a set of $k$ orthogonal matrices in $\mathbb{R}^{n \times n}$ with maximal summed pairwise distance (in terms of L2 operator norm). Any ideas? I am thinking of just doing ...
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32 views

Prove that a set is orthonormal on $L_2$

I would like to prove that the set of elements: \begin{equation} A_n(t)=\left\{\frac{1}{\sqrt{2\pi}}e^{int}\right\}_{n=-\infty}^{\infty} \end{equation} is an infinite orthonormal set, on space ...
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Show that an ONS $\{\phi_n\}$ is complete in $L^2(I)$ iff $\sum_{n=1}^\infty(\int_{[a,x]}\phi_n)^2=x-a$ for all $x\in I$.

Suppose $I=[a,b]$ is an interval of the line. Show that an ONS $\{\phi_n\}$ is complete in $L^2(I)$ iff $\sum_{n=1}^\infty(\int_{[a,x]}\phi_n)^2=x-a$ for all $x\in I$. My Work: If we suppose ...
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1answer
25 views

Planes And Lines

Given :Point $A(1,2,4)$ and plane $P: x-y+z+2=0$ How to find coordinates of point $A'$ the symmetric of point $A$ with respect to plane $P$
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Question about projecting vector onto Subspace

Consider subspace $S \in R^4$, spanned by the vectors: $v_1 = (1,0,-1,1)^T$ and $v_2 = (1,1,1,1)^T$. Let $v = (1,-1,-1,3)^T$. I want to find the projection of $v$ onto $S$; that is, find the ...
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1answer
40 views

Determining if this is an orthonormal set.

$p(x)\:=\:a+ax+ax^2$ and $q(x)\:=\:b+bx+bx^2$ are vectors in $P_2$ The inner product is the dot product: $$\langle p,q\rangle=a_0b_0+a_1b_1+a_2b_2$$ the set is: ...
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1answer
28 views

Is there some clever way of finding out this SVD?

I need to find the singular value decomposition of $$\pmatrix{a&-b&0\\b&a&-b\\0&b&a}$$ I already determined sigma to be: $$\Sigma = ...
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1answer
22 views

Fundamental Subspaces: Orthonormal Bases

In matlab, we are asked to set A=rand(5,2)*rand(2,5) then to set Q=orth(A), w=null(A'), S=[Q W] the matrix S should be orthogonal. Why? (I have no clue on how to answer this)
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1answer
19 views

Why does $A=UDU^H$ = $\lambda_1 u_1 u_1^H + \lambda_2 u_2 u_2^H$?

Just trying to figure out why this is true: $$A=UDU^H \quad\Leftrightarrow\quad A= \lambda_1u_1u_1^H + \lambda_2u_2u_2^H$$ $U$ is a unitary matrix composed of the eigenvectors of hermitian matrix A. ...
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1answer
44 views

Orthonormal and/or Orthogonal Basis of a Pair of Vectors

I was hoping someone could verify if this is the correct way to answer this problem: Let $\mathbb{R^{2}}$ have the standard dot product. Classify the following pair of vectors as (i) basis, (ii) ...
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21 views

Why is $||\vec{x}^\perp||^2=0 \,\forall\, \vec{x}\in \mathbb{R}^n$?

Consider the decomposition of $\vec{x} \in \mathbb{R}^n$ where $\vec{x}=\text{proj}_V\vec{x}+\vec{x}^\perp$. From Pythagoras, $||\vec{x}||^2=||\text{proj}_V\vec{x}||^2+||\vec{x}^\perp||^2$. It ...
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3answers
44 views

Gram-Schmidt Process and Orthogonal Components

Let the Gram-Schmidt process transform the vector system $(a_{1}, ..., a_{n})$ into the system $(b_{1}, ..., b_{n})$. Show that the vector $b_{k}$ is the orthogonal component of the vector $a_{k}$ ...
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2answers
59 views

Find orthonormal basis of $\mathbb{R}^3$ with a given span of two basis vectors

What is an orthonormal basis of $\mathbb{R}^3$ such that $\text{span }(\vec{u_1},\vec{u_2})=\left\{\begin{bmatrix}1\\2\\3\end{bmatrix},\begin{bmatrix}1\\1\\-1\end{bmatrix}\right\}$? I was thinking I ...
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1answer
54 views

How does equality in Bessel's Inequality prove an orthonormal complete sequence?

I've been searching around for an answer to this question on the web for some time, but I keep coming up short (it may very well be that I don't have the right terms to be searching with). In any ...
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4answers
90 views

Eigenvalues and Eigenspaces of a Projection

Let $P$ be the orthogonal projection onto a subspace $E \subset V$ ($V$ being an inner product space) with $\mathrm{dim(V)}=n$, $\mathrm{dim(E)}=r$. Obtain the eigenvalues and eigenspaces, along with ...
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1answer
17 views

Proving orthonormality of system by sum of fourier coefficients

Let $f\in L^2(\mathbb R)$. Prove the system $\{f(t-n)\}_{n\in\mathbb{Z}}$ is orthonormal if and only if $$\sum_{k\in\mathbb{Z}}|\hat{f}(\omega+2\pi k)|^2\equiv 1$$ I have no clue how to prove ...
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1answer
33 views

Among all unit vectors $\vec{u}=\scriptsize\begin{bmatrix}x\\y\\z\end{bmatrix} \in \mathbb{R}^3$ which one minimizes the sum $x+2y+3z$?

My first instinct, of course, was to use Lagrange multipliers, but I have to use linear algebra to solve this. How would I construct an orthonormal basis in this case? I'm not sure how to ...
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1answer
30 views

Finding a vector for an orthonormal basis after using gram-schmidt.

So I started with a basis for $P_3$ (polynomial of degree less than 3). $$\{1,x,x^2\}$$ which has inner product define as $$\langle p,q \rangle=p(-1)q(-1)+p(0)q(0)+p(1)q(1)$$ For this product I found ...
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1answer
89 views

Gram-Schmidt and Inner Product Spaces

Consider the matrix $A=\begin{pmatrix}2&1\\1&2\end{pmatrix}$. We define a new inner product over $\mathbb{R}^2$ given by $\langle\vec{v},\vec{w}\rangle=\vec{v}^T\cdot A\cdot \vec{w}$. Find ...
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2answers
37 views

Does $\langle Fv_j, Fv_j \rangle = \langle F^*v_j, F^*v_j \rangle$ imply that $F$ is normal?

Let $\{v_1, \ldots, v_n\}$ be an orthonormal basis of an inner product space $V$ and a linear $F:V\to V$ s.t. for all $j$: $$\langle Fv_j, Fv_j \rangle = \langle F^*v_j, F^*v_j \rangle.$$ Does it ...
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2answers
97 views

If $A$ is orthogonal and symmetric, there is an orthonormal basis consisting of eigenvectors of $A$

Let $A ∈ M(n, \mathbb{R})$ be an orthogonal symmetric matrix. Show that $\mathbb{R}^n$ has an orthonormal basis consisting of eigenvectors of $A$. What happens if $A$ is assumed to be only ...
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0answers
70 views

Eigen Decomposition

I have this expression, $W= U^TDU$ where $D$ is a diagonal matrix and U consists of orthonormal columns. $U$ and $D$ are not in the same dimensions. Say $U$ is $n\times m$ and $D$ is $n \times n$. ...
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1answer
16 views

Orthonormality of $\{1,\cos nt,\sin (n-a)t\}_{n\in\mathbb N}$.

We know that the system $$\{1,\cos nt,\sin nt\}_{n\in\mathbb N}$$ is orthonormal on $[0,2\pi]$ respect to $\frac{1}{\pi}\int_{0}^{2\pi} \cdot\ dt$. My question is: For what values of the parameter ...
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1answer
56 views

Big error in basis of tensor product space

Sorry I am currently somehow confused by the following: The Legendre polynomials $(P_l)$ form an ONB of $L^2(0,\pi)$ and the complex exponentials $(\frac{1}{\sqrt{2\pi}}e^{in \theta})_n$ form an ONB ...