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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Proving an orthonormal set is an orthonormal basis in Hilbert space [duplicate]

Consider a separable Hilbert space $H$, and $\{g_n\}$ is an orthonormal basis of $H$. Now there is an orthonormal set $\{f_n\}$ that satisfies $\sum_n\|f_n-g_n\|^2<1$. Show that $\{f_n\}$ is also ...
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Question concerning cross product and orthonormal vectors

Assume we have a vector $u= (u_1.u_2, u_3) \in R^3$ My problem is to find vectors $\vec w, \vec v$ such that $u= v \times w$ All vectors should be orthonormal. If $u= (u_1, u_2, u_3)$ ,is there a ...
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31 views

$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$, $|c_{k}| \leq \frac{1}{k}\}$ is compact

Let $H$ be a complex inner product space that is also a complete metric space with respect to the distance induced by the inner product. Assume $\{u_{k}\}_{k=1}^{\infty}$ be an orthonormal set in ...
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21 views

Linear Alegbra - Orthonormal Basis

Let $v_1=(1,2,2)$, $v_2=(1,0,1)$ in $R^3$ Find an orthonormal basis $\{u_1,u_2\}$ to $Sp\{v_1,v_2\}$ Find vector $w \in Sp\{v_1,v_2\}$ so $w \bullet u_1=3$ and $w \bullet u_2=9$ My solution I'm ...
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Separable dual space implies existence of complete series

Let $\{x_n\}$ be a basis for a Banach space $X$ and let $\{f_n\}$ be the associated sequence of coefficient functionals. Prove or disprove: if $X^\ast$ is separable, then ${f_n}$ is complete ...
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Proving that there is an inner product such that a linearly independent subset of an inner product space is an orthonormal basis

I encountered a question that asked me to proof that if $V$ is a real inner product space and there is a subset $\{v_1, \dots, v_n\}$ of linearly independent vectors in $V$, then there exist an inner ...
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Orthonormal Basis for $[5,1+t]^{\perp}$

Consider the vector space $\Bbb{V}=P_3(\Bbb{R})$ of the real polynomials of degree less or equal 3, with the inner product given by $$\langle f,g\rangle=\int_0^1f(t)g(t)dt,\forall f,g\in\Bbb{V}.$$ ...
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81 views

A question about integral-squared error.

We consider the problem of representing a time function, or signal, $x(t)$ on a $T$-s interval $(t_0, t_0+T)$, as an expansion. Thus we consider a set of time functions $\phi_1 (t), \phi_2(t), ..., ...
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What is the orthonormal basis for constant polynomial?

Hi I am trying to solve a question about projection of f onto subspace. Given information is f(x)=1/x and inner product is interal of f(x)g(x) on [1.3]. And i have to show that nearest constant ...
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Orthogonal Complement

"Let $\Bbb{V}$ be a vector space with an inner product $<\cdot,\cdot>$, and $S\subset\Bbb{V}$. We define the orthogonal complement of $S$, denoted by $S^{\perp}$, as follows: ...
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Identities for the Hilbert–Schmidt norm of products of projections.

I've been studying different metrics on the Grassmannian $Gr(k,n)$ of k-dimensional linear subspaces of $\mathbb{R}^n$ and found myself needing some identities for the norm of a product of orthogonal ...
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29 views

Why are non-polynomial Legendre functions and Legendre polynomials not orthogonal?

The eigenfunctions of distinct eigenvalues for a Hermitian operator are proved to be orthogonal. Why does the same not apply to Legendre polynomials and functions that have different eigenvalues ? ...
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31 views

Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.

Let $V$ be a finite-dimensional inner product space over field $F$, and let U be a subspace of $V$ . Prove that the orthogonal complement $U_{\perp}$ of $U$ with respect to the inner product $\langle ...
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Clarification on Some Definition of Inner Product Space

Suppose $V$ is finite-dimensional Real vector space and $T\in \mathcal{L}(V)$. Suppose that $V$ has a basis $(e_1,e_2,\ldots, e_n)$ of eigenvectors of $T$, every element of $V$ can be written as a ...
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Ortonormal basis of unitary operator and its spectral decomposition - check my solution.

Dear fellow mathematicians, I'm trying to do a linear algebra exercise, but I have no idea whether I have a correct plan of solution. Here is the problem: Find orthonormal eigenbasis (not sure if ...
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Gram-Schmidt-procedure with PARI/GP

Can the Gram-Schmidt-procedure to find an orthogonal basis of a vector space spanned by given linear independent vectors be easily done in PARI /GP or do I have to program the procedure ?
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Finding quaternion that transforms to particular basis

I want to find a quaternion $x \in{\mathbb{H}} $ that transforms (rotates) the $ i,j,k $ basis to a particular basis. In equations: $$ x i x^{-1} = a_1 $$ $$ x j x^{-1} = a_2 $$ $$ x k x^{-1} = a_3 ...
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Prove $\ell_{ki}\ell_{kj}=\delta_{ij}$

Prove $\ell_{ki}\ell_{kj}=\delta_{ij}$ where $\{\hat{\mathbf{e}}_i\}$ and $\{\hat{\mathbf{e}}_i'\}$ are sets of orthonormal basis vectors for $i\in\{1,2,3\}$, $\ell$'s are the direction cosines ...
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35 views

Linear transformations in bilinear form

Be $f:V \times V \to F$ a bilinear pattern and $V$ of finite dimension. Is it correct that for every linear transformation $T:V \to V$ exists another linear transformation $T':V \to V$ for which: ...
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How can I prove that the span of an a subspace and it's orthogonal complement is the whole vector space?

The book Linear and Geometric Algebra explains the following theorem in a way that I haven't been able to figure out: If $\mathbf{A}$ and $\mathbf{B}$ are subspaces of a vector space $\mathbf{B}$ ...
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Normalize matrix so that its projection equals identity - which to use?

I wish to normalize a given matrix M, n by k, so that its projection matrix equals the identity. This is to speed up computations. The transformation/decomposition would look like this: Generate M by ...
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37 views

Eigen vectors of the matrix whose columns are eigen vectors of the original matrix

Consider a matrix $A$ of dimension $n$X$n$ whose eigen vectors are $y_1,y_2,y_3,...,y_n$ and are linearly independent. What are the properties of the eigen vectors of the matrix $P$ whose columns are ...
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How to find orthonormal basis for inner product space?

In $\mathbb{R}^3$ we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$ How can I find an ...
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A problem on orthonormality of a set of complex functions

The following is a problem of an undergraduate exam test:
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51 views

Fourier coefficients with respect to an orthonormal basis for an inner product space

$V = \operatorname{span}(S)$, where $S = \{(1, i, 0), (1 - i, 2, 4i)\}$, and $x = (3 + i, 4i, -4)$. Apply the Gram–Schmidt process to the given subset $S$ of the inner product space $V$ ...
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139 views

Showing a certain sequence is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$

The problem is to show $$\left\{\frac{1}{\pi^{1/2}(i+z)}\left(\frac{i-z}{i+z}\right)^n\right\}_{n=1}^{\infty}$$ is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$ In another exercise, I have ...
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103 views

Project sin(x) onto orthonormal basis that span ${(1, x, x^2, x^3, x^4, x^5)}$ on domain $[-\pi, \pi]$

I am self-studying LA through Linear Algebra Done Right 2nd ed. I probably made a blatant error somewhere but I have been stuck for a whole day now. The book gave the answer $0.987862x − 0.155271x^3 ...
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Existence of periodic orthogonal basis in $L^2([0,1])$ which is not trigonometric?

Let $$ \psi(x) := \sin(\pi x). $$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$. My question is the following: Are there other examples ...
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Gram-Schmidt: Do the sets have some sort of order?

I'm learning about the Gram-Schmidt process: I have some subspace basis $A$ with three vectors: $$A = \{a_1,a_2,a_3\}.$$ Based on it, we will create an orthonormal basis $B$ with three vectors, ...
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Calculating an orthonormal base given another base.

Let $$W = \operatorname{span}(\{(1,1,1),(0,0,1)\})$$ Find an orthonormal base $B$ of $W$. So. An orthonormal set is a base whose elements are orthogonal with each other and their length is ...
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60 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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57 views

Orthonormal Basis of $L^{2}(0,1)$?

Are the functions $e_n := e^{i\cdot(2n+1)\cdot\pi\cdot x}$, $n \in \mathbb{Z}$ an orthonormal basis of $L^{2}(0,1)$? I suppose it is true, but I haven't been able to prove it myself yet.
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Orthonormals, v1,v2.

Given: $2$ vectors $v1,v2 \ne 0$ and $v1 \ne v2$ and $v1,v2 \in R^n$ Prove: $\{v1\}^\bot = \{v2\}^\bot$ if and only if ${v_1,v_2}$ are linearly dependent. Well, I do have a solution for this but I ...
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“Orthonormal” parameterization of solid sphere?

The standard parameterization of the solid sphere of radius $r$ centered at the origin in $3$-space is ...
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Given a symmetric matrix A, find an orthogonal matrix S such that $S^TAS$ is a diagonal matrix

Given the symmetric matrix: $$A = \left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right)$$ find an orthogonal matrix $S$ such that $S^TAS$ is a ...
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Orthogonals,Orthonormals,bases.

Given: $B = ((2/3, 1/3,2/3),(1/3,2/3,-2/3),(2/3,-2/3,-1/3))$ How can I check if B is an orthonormal base of $R^3$? I thought about first checking if these vectors are a base. Then checking if they ...
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Does R' = R*R'*R^(-1) hold?

Given two 3D rotation matrices ($3\times 3$) $R$ and $R'$, does this equivalence hold?: $R' = R*R'*R^{-1}$ My intuition tells me so, but I can't find a formal proof for it. Thanks.
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151 views

Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
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Sturm-Liouville(-like) differential equation

I encountered the following Sturm-Liouville-like (SL-like) partial differential equation in a project, and I decided to solve them numerically ($' = \frac{d}{dx}$): $(x(x+1) y’)’ – l(l+1) y = A ...
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Nonseperable Hilbert space: Explicit ONB?

Every Hilbert space admits an ONB by axiom of choice. For separable Hilbert spaces this can in fact be constructed by Gram-Schmidt. For nonseparable Hilbert spaces there can be no general construction ...
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Orthonormal matrix by diagonal matrix by orthonormal matrix transpose. [closed]

Let $D$ be a diagonal $n \times n$ matrix and $V$ an $n\times n$ matrix whose columns are orthonormal. Is it true that $$VDV^T = D?$$ If so, why?
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Two orthonormal vectors in space with finite decimal representation

I'm trying to formulate an exercise related to linear algebra, and for that I need two vectors in $\mathbb{R^3}$ which have unit length, are orthogonal to one another, don't have any zero ...
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Find the orthogonal projection of the vector… Help appreciated!

Does anyone know how I would go about answering question (b)? (b) Find the orthogonal projection of the vector u = (2,-1) on the vector v = (3,-2). http://i.imgur.com/nWcotnQ.png
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Orthonormal set problem

A)For the First three member of $(x_0, x_1, x_2, ... ) $ with respect to $$x_{j}(t)=t^{j}$$ in $[-1,1]$ , use the inner product function below to make them orthonormal. $$\langle x,y\rangle ...
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Want to show that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1-periodic $L^2$ functions.

So here is my problem, I would like to prove that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1- periodic $L^2([0,1])$ functions with respect to, $$\langle ...
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33 views

Orthonormal bases with canonical basis

Consider $\mathbb{R}^3$ with two orthonormal bases: the canonical basis $e = (e_1,e_2,e_3)$ and the basis $f = (f_1, f_2, f_3)$, where $f_1 = \sqrt{3}(1,1,1), f_2 = \sqrt{6}(1,−2,1), f_3 = ...
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37 views

Enlarging the space $PC(a,b)$ to include functions with one or more infinite singularities

I'm reading a Fourier analysis book and on the chapter about convergence and completeness of orthogonal sets of functions I have one part which I don't understand. I have uploaded the part as an image ...
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29 views

Orthonormal basis, decompse then add back

This is kind of a stupid question and I am taking some risk of getting some down-votes here, but, I can't resist posting it. Suppose $(u_1, u_2)$ is an orthonormal basis for $R^2$, and let $x$ be an ...
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99 views

Inner product space, orthonormal bases and change of basis.

I define unitary as $B*B=I$ I know that part (i) requires me to show the matrix coefficients are that of the inner product for bases A and B, however I am unsure how to get to this. Any help would ...
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44 views

Find the orthogonal complement of W

Consider the finite inner product space $\mathbb{R}^3$ where the inner product is the normal dot product. Find the orthogonal complement of W where W is $W = {(x,y,x+y) : x,y \in \mathbb{R}} \subset ...