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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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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58 views

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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29 views

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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40 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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39 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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32 views

“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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24 views

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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44 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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24 views

Hilbert space (nonseparable): 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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60 views

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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31 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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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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28 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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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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33 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 ...
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Construct an orthonormal basis from another set of basis?

Consider the following bais: $\phi_n(x) = \sum^N_{i=0} d^i_n u_i(x)\:\:\:\:\:\: (1)$ where $\phi_n(x)$ is some set of basis that must obey: $\int_\Omega \phi_n(x) \phi_m(x) dx = ...
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24 views

How to prove perpendicular vectors problem

If I have: $$|\underline a + \underline b| = |\underline a - \underline b|$$ how do I prove that $\underline a$ is perpendicular to $\underline b$?
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Finding an orthonormal basis from an existing one in a Hilbert space

Suppose we are given a separable Hilbert space $H$ with countable orthonormal basis $\{e_n\}$. Suppose we are given an orthonormal set $\{f_n\}$ such that $\sum\|e_n-f_n\| < 1$. How do we prove ...
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Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
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Question about maximal orthonormal subset in infinite dimensional vector space

The question is this: Let $A$ be an orthonormal subset of vectors in an infinite dimensional vector space $V$. Suppose for every $0\neq y \in V-A$ there is a $v \in A$ so that $\langle v,y ...
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35 views

Orthonormal linear combination

Let $(a_k)_{k\geq 1}$ a bounded sequence, $(e_n)_{n\geq 1}$ an orthonormal basis of an Hilbert space $H$. Define $x_n:=\frac{1}{n}\sum_{k=1}^na_ke_k$ Then $x_n\rightarrow0$ I proved that ...
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Rows of orthogonal matrix from an orthonormal basis of $R^n$

The question is: Let $U$ be an $n \times n$ orthogonal matrix. Show that the rows of U form an orthonormal basis of $\mathbb R ^n $. So far I have stated: Since $U$ is orthogonal its column vectors ...
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Proving that the function set $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set

I have the the following problem from my Fourier analysis book: Show that $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set in $PC(0,l)$, i.e. class of piecewise ...
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Orthonormal zero Function

I have this exercise Let H be a Hilbert space with orthonormal basis $\{e_n | n\in N\}$ and let $f_n = e_n + e_{n+1}$ If $\langle f,f_n \rangle = 0$ for all $n$ how do I prove that $f=0$ I think i ...
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Orthonormal basis of $\mathbb{C}^{r}$

I have to prove that $\{f_1,f_2,..,f_r\}$ is an orthonormal basis of $\mathbb{C}^{r}$ where: $f_j=\frac{1}{\sqrt{r}}(1,e^{2i\pi\frac{j-1}{r}},e^{4i\pi\frac{j-1}{r}},...,e^{2(r-1)i\pi\frac{j-1}{r}})$ ...
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53 views

Are These Orthonormal Functions?

Working through a text (self-learning, not homework), I reached this problem: Prove that the set of functions $\psi_n(x)=a^{-1/2}e^{i\pi nx/a}$ is orthonormal for integer $n$. So that's a fairly ...
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orthonormal vector properties

I have noticed a matrix property that is outlined below: I have a set of n orthonormal eigenvectors that form a basis in Rn. If these vectors are combined to form an nxn matrix where each column is ...
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64 views

Dot product with vector and its transpose?

I'm having trouble with the statement: $$||\textbf{v}||^2=\textbf{v}\cdot\textbf{v}=\textbf{v}^T\textbf{v}$$ taking $\textbf{v}$ as a column vector in an orthogonal matrix. How can you do the dot ...
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Orthonormalising a Basis $B = (b_1 ,…,b_4 )$ so that it contains $u_1 = \frac{b_1}{\sqrt{\langle b_1,b_1\rangle }}$

I have just a small question: I have a scalar product and a Basis $B = (b_1, b_2, b_3, b_4)$. The task is to Orthonormalize the Basis B so that it contains $u_1 = \frac{b_1}{\sqrt{\langle ...
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Generating orthonormal matrix

How can I generate matrix D described as follows: "D is a d*d rotation matrix with orthonormal unit vectors as columns. D is built by generating its lower triangular matrix independently from d(d-1)/2 ...
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82 views

Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
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1answer
34 views

Is every unit vector in $\mathbb{Q}^n$ the first column of a rational orthogonal matrix?

Equivalently, does every unit vector in $\mathbb{Q}^n$ belong to some orthonormal basis for $\mathbb{Q}^n$? This is clearly true for $\mathbb{Q}^2$, and for $\mathbb{Q}^3$ it seems to be true for ...
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Diagonal Transformation

Given an orthogonal matrix $r \in SO(n,\Bbb R)$), and some diagonal matrix $d$ of the same size $n \times n$; Is the matrix $r\cdot d\cdot r^{-1}$ also diagonal?
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Prove of inner product space and orthonormal system's necessary condition to be complete

I have no idea how to start, anything would help, thank you!
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16 views

Identification of tensor product spaces

Let $P_1, P_2$ be (probability) measures, $\Omega_1, \Omega_2 \subset \mathbb{R}^n$ . Prove that $L_{P_1 \otimes P_2}^2(\Omega_1 \times \Omega_2)$ and $L_{P_1}^2(\Omega_1) \otimes L_{P_2}^2(\Omega_2)$ ...
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28 views

Show that $T$ is a linear transformation given Orthonormal basis

Suppose that $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and suppose that $\{v_1,v_2,\cdots,v_n\}$ and $\{Tv_1,Tv_2,\cdots,Tv_n\}$ are orthonormal basis of $\mathbb{R}^n$. Prove that $T$ is a linear ...
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44 views

Linear Algebra Quick Question on orthonormal basis and inner product

I have a question asking to find an orthonormal basis of $p_2$ with respect to the inner product =2 X integral from 0 to 1 p(x)q(x)dx. What do I do with the 2 in front of the integral? When I solve ...
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58 views

Finding an orthonormal basis for the space $P_2$ with respect to a given inner product

I am so confused on what to do for this question. The questions asks to find an orthonormal basis of $P_2$, the space of quadratic polynomials, with respect to the inner product $$ \langle p, ...
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45 views

Name for multiples of orthogonal matrices

Is there a name for a matrix which is a multiple of an orthogonal matrix? I.e. a square matrix $A$ which satisfies the condition $$A^TA = AA^T = \lambda I$$ where $\lambda$ is some scalar (which ...
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42 views

Finding the dimension and basis of an orthogonal space

I am trying to find the basis and dimensions of the the space orthogonal $S$ which is in $\mathbb R^3$. $$S = \begin{bmatrix}1\\2\\3\end{bmatrix}$$ So the dimension would be two because it is $3 - ...
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3answers
119 views

Understanding the Gram-Schmidt process

I would like to better understand the gram-schmidt process. The statement of the theorem in my textbook is the following: \noindent The Gram-Schmidt sequence $[u_1, u_2,\ldots]$ has the property that ...