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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Gram-Schmidt Process, finding orthonormal basis

Suppose I'm given $2$ random vectors $(v_1,v_2)$ I want to find orthonormal basis $(w_1,w_2)$ Are the following equivalent? for the $w_2$ case $$w_1=\frac{v_1}{\|v_1\|}$$ ...
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4answers
43 views

How do I find orthonormal basis of $U$?

Let $U$ be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find orthonormal basis of $U$?
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2answers
31 views

Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis for some inner product.

Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis with respect to some inner product. How would i go about doing this?
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16 views

Point in $U$ closest to $(2,0,1,6)$

Let $U$ be the intersection of $x_1+x_2+x_3+x_4=0$ and $x_1+2x_2+3x_3+4x_4=0$ in $\mathbb{R}^4$. The basis of $U$ is $((1,-2,1,0), (2,-3,0,1))$ and the basis of the orthogonal space $U^\perp$ is ...
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2answers
38 views

Write a vector as a linear combination of orthonormal set vectors

How can I determine whether a vector can be expressed as a linear combination of a orthonormal set vectors ? Let's say I have a orthonormal set of vectors $\{v_1, v_2\}$: $$ ...
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1answer
19 views

Orthonormal Basis and Orthonormal Set

I was pretty confused about these two terms: 'orthonormal basis' and 'orthonormal set'. I know the orthonormal basis is just the normalized vectors of the orthogonal basis, but what about the ...
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2answers
18 views

summing inner product of orthonomal basis

I need some help with some very basic linear algebra when doing calculations in inner product space. Here is a line I got lost when reading a book... \begin{align*} ...
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1answer
30 views

Eigenvalues and Eigenvectors relating to orthogonal basis and diagonal matrices

Find the eigenvalues and eigenvectors of the matrix. $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & -1\\ 0 & -1 & 1 \end{bmatrix}$$ As we have seen in the ...
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0answers
37 views

Find the best approximation of $y = e^x$ in $P_2 [-1,1]$

$L_2$ norm weighted by $\omega$(x) = $\frac{1}{\sqrt(1-x^2)}$ The least-squares approximating polynomial Pn(x) of f(x) using Chebyshev polynomials is given by: $P_n(x) = a_0T_0(x) + a_1T_1(x) + · · ...
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2answers
67 views

Orthonormal Basis Question: Linear Algebra

I've been staring at this proof for a long time so any suggestions would be of great help! Prove that for any $m\times n$ matrix $A$ there is an orthonormal Basis $B =\{ v_1,\ldots,v_n\}$ of $\mathbb ...
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2answers
46 views

Proof of claim on orthonormal elements in an inner product space

Let $X$ be an inner product space and $\{e_{n}\}_{n=1}^{\infty} \subset X$ be an orthonormal set. Show that $$ \sum_{n=1}^{\infty}|\langle x,e_{n}\rangle\langle y, e_{n}\rangle| \leqslant ...
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2answers
16 views

Proving a basis for inner product space V when $||e_j-v_j||< \frac{1}{\sqrt{n}}$.

Suppose $(e_1,e_2,...,e_n)$ is an orthonormal basis of the inner product space $V$ and $v_1,v_2,...,v_n$ are vectors of $V$ such that $$||e_j-v_j||< \frac{1}{\sqrt{n}}$$ for each $j \in ...
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1answer
9 views

Lower bound of coefficients in an orthogonal decomposition

Is there a theorem of this form? Given a vector $x$ and an ONB $\{v_1, v_2, \ldots, v_n\}$, $\exists \, c>0$ such that $\langle x, v_j \rangle \ge c ||x||$ for some $j$. In other words, when you ...
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0answers
23 views

Describing the Unit Normal to a Cylinder in $\mathbb{R}^{3}$

This is problem 34.4 in Munkres' Analysis on Manifolds. Let $\mathcal{C} = \{ \ (x,y,z) \in \mathbb{R}^{3}\mid x^{2} + y^{2} = 1 \text{ and } 0 \leq z \leq 1 \ \}$. Orient $\mathcal{C}$ by declaring ...
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14 views

additional strucutre

Do I use the wrong calculations to find orthogonal projection?? We need to find a Projection of X onto V. In my calculations I am getting the same result as in the textbook but multiplied by factor of ...
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1answer
24 views

Show that $A$ is orthogonal iff the column vectors of $A$ form an orthonormal basis

My book says that: (a) $A$ is orthogonal if and only if the row vectors of $A$ form an orthonormal basis of $\mathbb{R}^n$ under the euclidean inner product; and (b) $A$ is orthogonal if and only ...
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1answer
41 views

Finding an orthonormal basis with only one vector

I'm programming a little thing to project things to my screen from a variable amount dimensions, and when I tried to implement the ability to move the origin and rotate the direction looked in strange ...
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2answers
53 views

Gram Schmidt and Inner Product

I'm working on this problem: Show that if $(e_1...e_n)$ is an orthonormal basis constructed using the Gram Schmidt process from $(v_1...v_n)$, then for any $j,$ $\langle e_j, v_j \rangle > 0$. ...
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28 views

Is this geometric Interpretation of $Q^T$ being orthonormal if $Q$ is orthonormal valid?

I was reading the book - Linear Algebra and its Applications, when I saw - Remark 2. Since $Q^T = Q^{-1}$, we also have $QQ^T = I$. When Q comes before $Q^T$, multiplication takes the inner ...
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34 views

Finding if the orthonormal basis of P1 with a given inner product

Hi, So I know for a basis to be orthonormal, the inner product needs to be 0 between components. So the standard basis <1,t> would not work in this case, as the integral from 1 to zero of t is ...
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0answers
38 views

orthonormal basis in $L^2$ space

Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty $ is an orthonormal basis for $L^2 (S \times S)$. Thanks ...
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1answer
7 views

A list is a basis if norm of difference with orthonormal basis is bounded by given constant less than unity

This is a problem from Linear Algebra Done Right, 3rd Edition, problem 14 in 6.B. Given $\{e_1, \dotsc, e_n\}$ is an orthonormal basis of $V$, and $||v_j -e_j|| < \frac{1}{\sqrt{n}} \; \forall \; ...
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1answer
26 views

SVD of (2,1,-2) not ok

I'm trying to find the SVD of $$ \begin{pmatrix} 2&1&-2\\ \end{pmatrix} $$ I found $$\Sigma , u$$ But on the V matrix I got $$ \begin{pmatrix} ...
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3answers
46 views

Prove that $C^nx = \frac{1}{2^n}au + \frac{1}{3^n}bv$, for every $n \in N$

EDIT: I forgot to mention $C = 0.5uu^T + 0.33vv^T$ and now if I use it, I solve it easily. Given: $Cx = \frac{1}{2}au + \frac{1}{3}bv$, $x \in R^2$, $u,v$ are orthonormal vectors in $R^2$, $x = au ...
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2answers
27 views

How does one write an transformation matrix under an orthonormal basis?

I have an orthonormal basis (consisting of either vectors or polynomials). How do I write a matrix for a linear transformation Tv=w for any vector v and w in the vector space V? Is there a general ...
2
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2answers
31 views

Unitary Matrix columns = Orthonormal Basis

So I'm trying to understand why the columns of a unitary matrix form an orthonormal basis. I know it has something to do with the inner product, but I don't fully understand that either (we learned ...
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33 views

Projection of inner product spanned by orthonormal basis

So the question asks: Consider the inner product $⟨f,g⟩=$$ \begin{align} \int_{-1}^{1} f(x)g(x) \ dx &\end{align}$$ $ on $P_2$, the space of all polynomials of degree 2 or less. Find the ...
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1answer
40 views

Normalize vector : $ \left(\frac{e^{\frac{-i\pi}{4}}}{\sqrt{10}}, -3\frac{e^{\frac{i\pi}{4}}}{\sqrt{10}} \right)$

This is a homework question that I am struggling with. I have to normalize the vector $$ \left(\frac{e^{\frac{-i\pi}{4}}}{\sqrt{10}}, -3\frac{e^{\frac{i\pi}{4}}}{\sqrt{10}}\right)$$ as I have to ...
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0answers
21 views

Jordan Normal Form of a self-adjoint Linear Transformation

Let $V$ a finite inner product space, $dim V = n \geq 3$. Let $w_1,w_2 \in V$ such that: $<w_1,w_2>=0$, $||w_1||=||w_2||=1$ where $||w||$ is the norm of a vector $w$. The inner product ...
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21 views

Orthonormal basis and vectors in a space V

Let's say I have an orthonormal basis of $V$ $(e_1,\ldots,e_n)$ and a bunch of vectors in $V$ $(v_1,\ldots,v_n)$ that may or may not be linearly independent. How can I then prove that those vectors ...
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0answers
22 views

Showing the Haar wavelet is a complete and orthonormal sequence within $L_2[0,1]$

I define the mother Haar wavelet to be: \begin{align} \phi(t) = \begin{cases} 1 &\mbox{if } 0 \leq t < 1/2 \\ -1 & \mbox{if } 1/2 \leq t \leq 1 \\ 0 &\mbox{otherwise}. ...
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1answer
30 views

Finding the Orthonormal Basis of a given Subspace

Hi, So I am not sure how to do this problem. I think what I have to do is find a basis for the space of 2x2 matrices that are orthogonal to the identity matrix s.t. the HS inner product. And from ...
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0answers
24 views

The direction of a normal vector

This is from my textbook I'm a little bit confused with the direction arrow of the normal vector, shouldn't it be the green vector?
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0answers
23 views

QR Factorization of a rank 1 real matrix

An answer in my textbook says that QR Factorization is possible for any real matrix, but doesn't one need to have a full column rank in order to do this (through Gram-Schmidt process), as this is what ...
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1answer
26 views

How to find orthonormal bases of a null space?

$\mathbf{H}$ is a $K\times N$ matrix of complex elements where $N>K$. I guess there must be some standard algorithms to find a set of orthonormal basis for its Kernel, i.e. set of all ...
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1answer
45 views

Finding the orthogonal projection of a given vector on the given subspace W of the inner product space V.

Let $V = R^3$ with the standard inner product $u = (2,1,3)$ and $W = \{(x,y,z) : x + 3y - 2z = 0\}$ I came up with the basis $\{(-3,1,0), (2,0,1)\}$ but these are not orthogonal to each other. I'm ...
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0answers
53 views

matrix optimization in diagonal and orthonormality constraint

I'd like to solve the following optimization problem. Y is a CxN matrix A is a CxC and diagonal matrix Q is a CxC and orthonormal matrix X is a CxN matrix $$ min_{A,Q} {|| Y − AQX ||}^2_F $$ The ...
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0answers
21 views

Orthonormal Basis of Hermitian matrices for a Hilbert Space of Operators

Consider a set of operators $O$ on a Hilbert space $V$ of dimension $d$. I could prove that $O$ is also a Hilbert space with dimension $d^2$ (inner product being $(A,B) = tr(A^\dagger B))$. Now I am ...
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1answer
49 views

Using direct sums, construct an inseparable Hilbert space with an uncountable orthonormal basis

Using direct sums, construct an inseparable Hilbert space with an uncountable orthonormal basis. This is Problem 13 in Chapter II in Reed & Simon, and I'm really stuck on this one. Would ...
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1answer
38 views

If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
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1answer
39 views

Prove that the sum of the modulus squared of $\hat{A}$ is independent of the complete orthonormal basis

Prove that the sum of the modulus squared of the matrix elements of a linear operator $\hat{A}$ is independent of the complete orthonormal basis used to represent the operator. I believe I know how ...
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1answer
27 views

Simultaneous diagonalization of two Riemmanian metrics

Let $M$ be a smooth manifold and $g,h$ two Riemannian metrics on $M.$ Can one find a local frame $(e_i)$ orthonormal with respect to $g$ such that $$ h(e_i,e_j)=0$$ $\forall i \neq j?$
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1answer
53 views

Gram-Schmidt Process to find an orthonormal basis for a matrix

By using the Gram-Schmidt Process find an orthonormal basis for the column space of the matrix: $$A=\begin{pmatrix}0 & -3 & 1 \\ 1 & 0 & 1 \\ 1 & -3 & ...
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65 views

Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting

I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
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0answers
26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...
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0answers
18 views

Orthonormal Basis and evaluation of expression

Let the vectors $\mathbf{a}$ and $\mathbf{b}$ be specified by $$\mathbf{a} = -3\mathbf{e_1} + 2\mathbf{e_2} - 8\mathbf{e_3}$$ $$\mathbf{b} = 5\mathbf{e_1} - 7\mathbf{e_2} + 3\mathbf{e_3}$$ where ...
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0answers
29 views

Normalizing disturbed rotation matrix

I was playing with simulations of Euler's equations of rotation in this question. This involves integrating an ordinary differential equation of a rotation matrix, $R$, which is calculated for all of ...
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0answers
19 views

Coefficients of Fourier-Bessel series for a Neumann condition

What is the expression for coefficients of Fourier-Bessel series for a Neumann condition? I know what it is for Dirichlet condition. $\frac{\partial f}{\partial x} = 0$
2
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1answer
29 views

Self Adjoint Linear Maps and Matrices with respect to orthonormal bases

Let $T: V \rightarrow V$ be a self adjoint linear map where $V$ is an inner product space. Known fact: With respect to any orthonormal basis- the the matrix for $T$ is conjugate symmetric. When ...
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1answer
13 views

Proof of Hesse Normal Form

Let $F$ be a hyperplane in $\mathbb{R}^n$. Let $0 \neq a = \begin{pmatrix} x_1 \dots x_n \end{pmatrix}^\top$ and $F=\{x\in \mathbb{R}^n \mid a \bullet x=0\}$. If we use $N=\frac{a}{\mid a \mid}$ we ...