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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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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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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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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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}, ...
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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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19 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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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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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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38 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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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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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
26 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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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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26 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), ...
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36 views

Orthogonal Matrix with a specific row

I have an assignment with the following question: Does an Orthogonal Matrix exist such that its first row consists of the following values: ($1$/$\sqrt{3}$, ...
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14 views

Orthonormal basis and representation of a vector with angle

Given two orthonormal vectors $u_1$ and $u_2$ which form a basis for $\mathbb{R}^2$. The vector $u \in \mathbb{R}^2$ can be represent as $$u = \cos \theta \ u_1 + \sin \theta \ u_2$$ where $\theta$ is ...
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69 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
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Real affine variety of $d$ orthonormal vectors in $\mathbb R^n$

I'm interested in the affine variety $$ V = \left\{ \, A\in \mathbb R^{d\,\times\, n} \, \middle| \, A\,A^T = I \, \right\} \subseteq \mathbb R^{d\, \times\, n}, $$ where $n\ge d$ and $I$ is the ...
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orthonormal sequence in $L^2[0,1]$ - how to prove these following equivalent terms?

I've been asked this following very interesting question and would like some help figuring out why it is true :) Let $u_n$ be an orthonormal sequence in $L^2[0,1]$ Prove that the following are ...
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Inner product exterior algebra

I have to prove that if V is a real vectorial space (dimV=n) with inner product (.,.) then if we define $$ (v_{1}\wedge v_{2}\wedge...\wedge v_{k},w_{1}\wedge w_{2}\wedge...\wedge w_{k}) ...
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33 views

Using my orthonormal basis to find a polynomial that best approximates the polynomial $t^3$

I want to find the second-order polynomial that best approximates $t^3$, with respect to the norm of the vector space $V$. I first proved the bracketing map given in the problem was indeed an inner ...
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A basis $B$ is orthonormal if and only if $\langle f,g \rangle=[f]_B\cdot [g]_B$ for all $f$ and $g$ in $V$

Consider a finite-dimensional inner product space $V$. If $B$ is a basis for $V$, show that $B$ is orthonormal if and only if $\langle f,g \rangle=[f]_B\cdot [g]_B$ for all $f$ and $g$ in $V$. I'm ...
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30 views

Fill in the missing entries of $Q$ to make $Q$ an orthogonal matrix?

I need to fill in the missing entries of $Q$ to make $Q$ an orthogonal matrix. I have no idea how to solve this problem out. I was hoping for some hints on how to go about this. Problem: $$Q= ...
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35 views

Need help with question regarding orthonormal basis

Let v1=(3/√14, 1/√14, -2/√14), v2= (1/√3,-1/√3,1/√3), v3=(1/√42,5/√42,4/√42) and its also known that the set {v1,v2,v3} is an orthonormal basis of R3. Write the vector u=(100,-200,300) as a linear ...
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Matrix of orthogonalprojection

whats the matrix of an orthogonalprojection? This question was part of a longer task: Step: two vectors were given: $v_1 = (i, 0, 1)$ $v_2 = (0,i,1)$ U is a vector space, spanned by v1 and v2 ...
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42 views

Determine whether the given the orthogonal matrix represents a roation or reflextion…?

I am given the matrix $$ \begin{bmatrix} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2\\ \end{bmatrix} $$ I think this is a reflection because I tied sketching a ...
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Symmetric matrix if symmetric linear transformation

I want to proof the following theorem: With respect to any orthonormal basis, if the 2 $\times$ 2 matrix $\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ represents a ...
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35 views

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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$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$, $|c_{k}| \leq \frac{1}{k}\}$ 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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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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26 views

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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85 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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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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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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40 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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84 views

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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78 views

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 ...