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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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
36 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
26 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
18 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
17 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
22 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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0answers
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
24 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
50 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
20 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
53 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
16 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
26 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
27 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
78 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
34 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
91 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
63 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
14 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
52 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 ...
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1answer
37 views

Showing an Operator is Well-Defined and Bounded

Let $\{e_n\}_{n \in \mathbb{N}}$ be an orthonormal system within $\ell^2$. Fix a sequence $\lambda = (\lambda_1, \ldots , \lambda_n , \ldots) \in \ell^{\infty}$ and define $ \displaystyle Tf = ...
4
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1answer
82 views

Interchange summation and differentiation for ONB

Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$ Now assume we have that $$\frac{d}{dx}e_n = \lambda_n e_n.$$ Assume $f \in H^1[0,1],$ so i.e. $||f'||_{L^2} < \infty$ ...
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1answer
31 views

$\langle f, \phi_n \rangle = 0 \implies f = 0$ is equivalent to the definition of orthonormal basis

Is there an "easy" way to see that if $\{\phi_n\}_{n=1}^\infty$ is a set of orthonormal functions in a Hilbert space then showing $\langle f, \phi_n \rangle = 0$ for all $n$ implies $f = 0$ is ...
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0answers
26 views

How to restrict an angle to a particular interval during iterative least squares inversion?

I have some data and I want to fit a model to it which is a function of 3 orthonormal eigenvectors, using damped iterative least squares inversion. My orthonormal eigenvectors A, B, and C are ...
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2answers
62 views

Do the statements hold in an inner product space over $\mathbb R$ as well?

Let $V$ be an $n$-dimensional inner product space over $\mathbb C$ and $f\in \mathcal L (V)$ normal. Show that: $f^2=f^3 \implies f=f^2 \implies f = f^*$ $f$ nilpotent $\implies f=0$ ...
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41 views

How can we recover $u$ from a Householder matrix?

Say, we have a Householder matrix $H$ (i.e. $H=H^T=H^{-1}$) and would like to find $u$ such that $$H=I-\frac{2}{u^Tu}uu^T.$$ Geometrically, it seems like a good idea to take some $x$ and try ...
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2answers
37 views

linear algebra - orthonormal basis [closed]

Determine an orthonormal basis {e1, e2, e3} such that e1 is parallel to the vector u = (1, 2, -1). Can you please help me find the ON-basis ?
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2answers
26 views

Determine whether set forms an orthonormal basis

Consider the three vectors $x^{1} = (1/\sqrt2, 0, −1/\sqrt2)^T$ , $x^2 = (0, 1, 0)^T$ , $x^3 = (1/\sqrt2, 0, 1/\sqrt2)^T$. Does the set $A = {x^1, x^2, x^3}$ form an orthonormal basis of ...
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0answers
20 views

See if vector set is basis of space using Gram Schmidt process

I have a problem my teacher gave me and I can't find an answer. She gave me a set of 3 vectors, $$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \begin{bmatrix} 7\\3\\1 \end{bmatrix} \begin{bmatrix} ...
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0answers
18 views

Orthogonality of continuous functions

Let $K(x,t)$ be a continuous kernel defined on $[0,1]\times [0,1]$ with an associated inverse kernel. Define a function $p(x,t)$ on $[0,1]\times [0,1]$ by $$p(x,t) = K(x,t) - \int_0^{t/2} ...
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1answer
50 views

Properties of Orthonormal Systems and Projections

Let $\{e_1, ... , e_n\}$ be a finite orthonormal system in an inner product space $(E, \langle \cdot , \cdot \rangle)$, let $F :=$ span$\{e_1, ... , e_n\}$ and let $P:E \to F$ be the orthogonal ...
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1answer
22 views

Rewrite an expression in terms of basis vectors

Given any vector k $\epsilon$ $R^{3}$ consider k= $\sum_{j=1}^{3}$ $c_{j}u_{j}$ where $u_{1}$,$u_{2}$,$u_{3}$ are the orthonormal basis vectors (I don't know how to make them bold sorry about that, ...
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0answers
40 views

In an infinite dimensional real inner-product space , can any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis?

Let $V$ be an infinite dimensional real inner-product space , then is it true that any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis ? Or at least is it true ...
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0answers
26 views

How to rewrite $PP^*$ in terms of sums of its column vectors?

$\begin{align} I & = & PP^* \\ & = & \begin{pmatrix} \bf{x_1} & \bf{x_2} &\dots & \bf{x_n} \end{pmatrix} \begin{pmatrix} \bf{x_1^*} \\ \bf{x_2^*} \\ \vdots \\ ...
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0answers
28 views

Give an example of a complete orthonormal sequence in L^2 ([ a, b]) for arbitrary a< b;

Give an example of a complete orthonormal sequence in L^2([ a, b]) for arbitrary a< b; I found an example of such But I can not prove its complet? Please advice?
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0answers
63 views

Find an orthonormal basis without Gram-Schmidt

Find an orthonormal basis for the subspace $ W = \text{span} \{(3, 0, 4, 0),(0, −2, 1, 0),(0, −3, 0, 1)\}$ of $\mathbb{R}^4$ Without using Gram-Schmidt process.
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0answers
35 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
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2answers
29 views

Why is the projection of a vector V onto a span W, independent of the orthogonal basis of W.

Very straightforward question. I have read time and again in my book that it is independent but I don't understand why? Wouldn't changing the basis mean changing the length of the projection?
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1answer
33 views

Find the orthonormal vectors $q_{1}, q_{2}, q_{3}$ such that $q_{1}, q_{2}$ span the column space of $A$?

We have given the matrix $$ A= \begin{pmatrix} 1 &1 \\ 2& -1 \\ -2 & 4 \end{pmatrix}$$ First the question asks find the orthonormal vectors $q_{1}, q_{2}, q_{3}$ such that $q_{1}, ...
3
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2answers
52 views

Orthonormal basis . Can I have more than one basis for the subspace?

Required to find an orthonormal basis for the following subspace of R4 I know that to find the othonormal basis, it is required that i find the basis for the subspace, then I use Gram Schmidt ...
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2answers
27 views

Find an orthogonal matrix that achieves a given vectorial transformation

Given a vector $\vec a\in\mathbb R^n$ and another $\alpha=(\|\vec a\|,0,\dots,0)$, how could I define an orthogonal matrix $M$ such that $M\vec a=\alpha$ and $M^{-1}=M^t$? For $\mathbb R^2$ I tried to ...
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1answer
39 views

Find an orthogonal basis for W.

Use the standard Euclidean inner product on $\mathrm R^4$. Let $W$ be the subspace of $\mathrm R^4$ spanned by $u_1 = (1, 1, 1, 1),$ $u_2 = (2, 4, 1, 5),$ $u_3 = (1, -5, 4, -8).$ Find an ...
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1answer
44 views

Orthonormal Hamel Basis is equivalent to finite dimension

Consider a Hilbert space which is infinite dimensional. If it is separable, it is well known that an orthonormal basis will be countable, while a hamel basis will be uncountable (since it is a ...
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1answer
69 views

Finding orthonormal basis for inner product in P2(C)

Let $p, q \in P_2(\Bbb C)$. Define the inner product by $$\langle p(x), q(x)\rangle = \int_0^1 p(x) \overline{q(x)} \, dx $$ How do I find orthonormal basis for inner product space?
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1answer
27 views

Find a point on a cercle in an orthonormed system given an angle.

In this inverted orthonormed system, I need to find a formula that gives the x and y coordinates of a point on a circle in function of an angle A that have its top on the center of the circle. I know ...
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1answer
22 views

Proofing the sum of the squre cosines of the angle between a vector and the vectors of the base.

I'm having trouble getting this proof right, its somthing like this Let $ \{e_1, e_2, ... e_n \}$ be an orthonormal basis, of a vector space with internal product $\mathbb V$, and $\theta_i$, the ...
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1answer
34 views

Eigenvectors, bilinear forms and orthonormal bases

I have calculated (a) to be $(1,-2,2)^t, (-2,1,2)^t, (2,2,1)^t$. For (b) I have made all of these of unit length ie taken 1/3 of each vector. I have verified these are orthonormal by checking ...
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1answer
28 views

Use the orthonormality of $u,v,w$ to write the following vectors as linear combinations of $u,v$ and $w$

Let $V$ be the vector space $\mathbb R^3$ with inner product $$(v,w)=3(v_1w_1)-2(v_1w_2)-2(v_2w_1)+5(v_2w_2)-3(v_2w_3)-3(v_3w_2)+3(v_3w_3)$$ where $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$. Part 1 ...
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1answer
30 views

Prove that the vectors u,v,w are orthonormal in V

Let V be the vector space R3 with inner product (v,w)=3(v1w1)-2(v1w2)-2(v2w1)+5(v2w2)-3(v2w3)-3(v3w2)+3(v3w3) where v=v1,v2,v3 and w=w1,w2,w3 Prove that the vectors u=(1,1,1), ...