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Questions tagged [gram-schmidt]

Questions relating to the Gram–Schmidt process, which takes a set of input vectors and produces an orthonormal set of vectors that spans the same subspace as the input set.

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Explain every steps of the Modified Gram–Schmidt algorithm

Can someone explain in details what every step in the modified gram Schmidt algorithm is doing? MGS algorithm This is what I think could someone correct me if I am wrong? We are using a series ...
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Where do I go from here? Using Gram-Schmidt Orthonormalization Process

The problem is Orthonormalize the following basis. The basis is ordered. Be sure to show your steps. B = {(3, 4, 0, 0), (-1, 1, 0, 0), (2, 1, 0, -1), (0, 1, 1, 0)} What I have attempted: I have: w1=...
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Show that $||v||^2 = ||P_0v||^2 + ||v - P_0v||^2$ for orthogoonal projection

I'm working on some practice problems from Noble & Daniel's Applied Linear Algebra (3rd), specifically here looking for help with question 5 from section 5.8 on pg. 232. Suppose that $P_0$ is the ...
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Gram-Schmidt orthogonalization process with specific dot product

I have three vectors $$v_1=(1,1,1)^T$$ $$v_2=(1,1,0)^T$$ $$v_3=(1,0,0)^T$$ and special dot product definition $$(\overline{(x_1,x_2,x_3)},\overline{(y_1,y_2,y_3)})=2x_1y_2+x_1y_1+2x_2y_1+x_3y_3$$ I ...
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Gram-Schmidt Procedure from “Linear Algebra Done Right”

The following content is from "Linear Algebra Done Right" book by Sheldon Axler, 6.31. There was a part of the proof what I don't understand is that $\begin{align*} \left\langle e_j, e_k\right\...
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Chebyshev Polynomials of the Second Kind from Orthogonality

I am tasked with finding the degree 5 Chebyshev-II polynomial, using the fact that it's orthogonal to those preceding it w.r.t the Chebyshev-II inner product. I am told to use the normalisation that ...
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Generation of Hermite polynomials with Gram-Schmidt procedure

I want to use the Gram-Schmidt procedure to generate the first three Hermite polynomials. Given the set of linearly independent vectors $\{1,x,x^2,...\}$ in the Hilbert space $L^2(R,e^{-x^2}dx)$, I ...
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Gram–Schmidt vs Modified Gram–Schmidt vs Stable Gram–Schmidt

I'm trying to understand what is the different between this 3 process, and till now what I understood is: Modify process add $r_{ij}=u^T_jw_i$ and $r_{jj}=\sqrt{\|{w_j}\|^2-\sum_{i=0}^{j-1} r^2_{ji}}$...
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Calculate an orthonormal basis

We have the following Gram-Schmidt algorithm: I want to calculate for the following vectors an orthonormal basis, with an accuracy of $\epsilon=5\cdot 10^{-3}$. \begin{equation*}a_1=\begin{pmatrix}...
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Choice of vectors from basis in Gram-Schmidt process

Say I have a basis for $\mathbb{c}^{2}$ composed of the vectors $(1,1), (4i,2i )$ with complex inner product. When I construct my orthogonal basis using the Gram-Schmidt process how do I make a choice ...
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Basis for orthogonal complement possibly with linear combinations

In $\mathbb{R}^4$, consider the subspace $W = Span(u_1, u_2,u_3)$ with $$u_1 = (-1,1,0,0)$$ $$u_2 = (-1,0,1,0)$$ $$u_3 = (-1,0,0,1)$$ $(a)$ Use the Gram-Schmidt Process to ...
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How do I make 3 given vectors with an unknown value $t$ in it, into an orthogonal set?

For what $t$ will the following vector be an orthogonal basis? \begin{align}u_1&= (1,t,t)\\ u_2&= (2t,t+1,2t-1)\\ u_3&= (2-2t,t-1,1)\end{align} Till now I have tried using the Gram-...
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Use the Gram-Schmidt Process to construct an orthogonal set of vectors using an inner product [closed]

This question on my Linear Algebra test has been stumping me. Every time I finish this problem I end up with vectors which aren't orthogonal. Use the Gram-Schmidt Process to construct an orthogonal ...
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Gram Schmidt procedure

For the subspace on the left in the image I am to find an orthogonal basis for the subspace. The answer is to the right in the image, and when I follow the procedure I get those vectors, however I ...
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Transformation matrix with respect to an orthonormal basis

I have this question here... Let $V$ be the span of $v_{1}=(0,1,2)$, $v_{2}=(-1,0,1)$ and $v_{3}=(-1,1,3)$. $(a)$Construct an orthonormal basis $B'$ for $V$ (usual dot product). $(b)$ ...
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Creating an orthonormal basis with Gram schmidt procedure error.

I have a question which says the following: Let $V$ be the span of $v_{1}=(0,1,2)$, $v_{2}=(-1,0,1)$ and $v_{3}=(-1,1,3)$. Construct an orthonormal basis $B'$ for $V$ (usual dot product). I ...
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Finding orthonormal basis of subspace?

Question I am completely lost on this problem. I know how to find it using Gram-Schmidt but I'm unsure of how to even find the subspace in this case, or how I would graph any of this. Is there another ...
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Upper Triangular Matrix and Gram-Schmidt

If $T$ is an operator on the finite dimensional vector space $V$ which has a basis $\mathcal{B}$ for which the matrix representation of $T$ with this basis is upper-triangular, does it follow that $T$ ...
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Gram-Schmidt orthogonality

Given a vector $v = (1,2,−1)$ in $\mathbb{R}^3$. Use the Gram-Schmidt process to find two vectors $v_1$ and $v_2$ such that $\{v,v_1,v_2\}$ is an orthogonal basis of $\mathbb{R}^3$.
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Abstract optimization (minimization problem)

I have some linear function $f(x) = ax + b$ and I am trying to find a polynomial $p \in \mathbb{P^3}$ such that $p(0),p'(0) = 0$ and $\int_0^1 |f(x) - p(x)|^2dx$ is minimized. This means $p = ax^3 + ...
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Question About Gram Schmidt Procedure

Let's say we have a basis A = {$\vec{a}_1, ..., \vec{a}_n$} for $\mathbb{R^n}$. If we apply the Gram Schmidt Procedure to A and get B={$\vec{b}_1, ..., \vec{b}_n$}, an orthogonal basis for $\mathbb{R^...
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Algorithm for orthogonalizing polynomials with specific inner product

I am attempting to generate a as big as possible collection of orthogonal polynomials $p_1, p_2, ..., p_n$, $\left\langle p_i, q_i\right \rangle = \delta_{ij}$ where the inner product is with respect ...
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Gram-Schmidt in $L^2[-1,1]$ with list of functions

Q: Apply Gram-Schmidt in $L^2[-1,1]$ to the list of functions $\{1,x,x^2,x^3\}$. You do not have to normalize. I have encountered Gram-Schmidt with vectors: $$U_1 = V_1$$ $$U_2 = V_2 - \frac{\left&...
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Classical Gram-Schmidt for matrix $A$

Let $$A=\begin{bmatrix}1 & 1 & 1\\ \epsilon & 0 & 0 \\ 0\ & \epsilon & 0 \\ 0 & 0 & \epsilon \end{bmatrix}.$$ On this page, this matrix $A$ is used to show the ...
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1answer
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Conditions for upper-triangular factor in QR factorization

Let $A,Q_1 \in \mathbb{R}^{m_1 \times n}$, $B \in \mathbb{R}^{m_2 \times n}$ $Q_2 \in \mathbb{R}^{(m_1+m_2) \times n}$, and $Q_3 \in \mathbb{R}^{ (n+m_2) \times n }$ with $Q_i$ having orthonormal ...
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Applying the Gram-Schmidt orthonormalization method in order to find an orthonormal basis for the subspace of $\mathbb{R}^4$

Apply the Gram-Schmidt orthonormalization method in order to find an orthonormal basis for the subspace of $\mathbb{R}^4$ that is generated by the vectors $(1, 0, 1, 1)^T , (1, 1, 0, 0)^T$ and $(0, 0, ...
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Existence of Orthonormal Basis of a Metric in a Manifold

Definition: A metric $ g$ on a manifold $ M$ is a tensor field of type $ (0,2)$ such that (1) it is symmetric, i.e. $ g(v,w)=g(w,v)$ for any $ w,v \in V_p, p\in M$, and (2) it is non-degenerate, i.e....
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Finding an orthonormal basis of the subspace using Gram-Schmidt method

Apply the Gram-Schmidt orthonormalization process to the vectors $[1,3,2]^T$ and $[1,0,1]^T$ in order to get an orthonormal basis of the subspace that they span. My Try: I took $u_1=[1,3,2]^T,u_2=[1,...
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Finding an orthonormal basis of the subspace

Find an orthonormal basis of the subspace: $$V = {[x, y, z, w]^T:x+y+z+w=0}$$ of $\mathbb{R}^4$ First I found a $4\times4$ determinant to verify whether they are non-singular or not. $$\begin{...
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Matlab Code-Include Iteration to QR Algorithm Gram-Schmidt - The Iterations of A will converge to Eigenvalues

Still need to add the iteration to the Matlab Code of the QR Algorithm using Gram-Schmidt to iterate until convergence as follows: I am having trouble completing the code to be able to iterate the ...
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Gram-Schmidt over GF$(2)$

I am reading the paper The Steganographic File System by Ross Anderson, Roger Needham, and Adi Shamir. On page 4, paragraph 2, the authors write: Finally, we use the Gram-Schmidt method to ...
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Typo in Gramm-Schmidt Orthonormalisation Process Proof?

The following is a proof from Appendix C (Linear Spaces Review) of Introduction to Laplace Transforms and Fourier Series, Second Edition, by Phil Dyke: Should the highlighted part be $v_i$ instead of ...
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Change of basis matrix from a basis to a orthonormal equivalent basis

Let $B_1 = \{v_1, v_2, v_3\} $ be a basis for $\mathbb R^3$ Let $B_2 = \{o_1, o_2, o_3\}$ be an orthonormal basis after executing the Gram-Schmidt algorithm on $B_1$ Let $$ P = \begin{pmatrix} ...
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Are there any cases where Modified Gram-Schmidt is more reliable than Householder for QR decomposition?

So, I am comparing different methods for finding the least square solution for $Ax=b$ where $$A(i,j)=\frac{1}{i+j+10}$$ for $i=1,2,...,10$ and $j=1,2,...,8$. Also $$b=[1,2,...,10]^T$$ I have made ...
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The Redundancy of applying the Gram-Schmidt Procedure on an orthogonal subset of $V$.

Is the following Proof Correct? For the facility of the reader the formal statement of the Gram-Schimdt Procedure is provided below. Proposition. If $\{w_1,w_2,\dots, w_n\}$ is an orthogonal set of ...
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Orthogonal bases for the eigenspaces.

I'm looking at a math problem which I have the solution for. The question looks like this: Question By hand, calculate the eigenvectors and find orthogonal bases for the eigenspaces. The ...
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deriving an orthogonal base for polynomials of degree 3

I am given the inner product space $\mathcal{P}_3$ with the ordered base 1, $X$, $X^2$. I have to apply the Gram-Schmidt process to derive an orthogonal base for $\mathcal{P}_3$. I got 1, $X$, $X^2$ - ...
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How to decompose a bivector into a sum of _orthogonal_ blades?

In Geometric Algebra, any bivector $B\in\Lambda^2\mathbb R^n$ is a sum of blades: $$B = B_1 + B_2 + \cdots$$ $$= \vec v_1\wedge\vec w_1 + \vec v_2\wedge\vec w_2 + \cdots$$ Each blade's component ...
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Why does Gram Schmidt preserve eigenvectorness?

Say you have 2 vectors in the same Eigenspace and apply GS to get two orthogonal vectors. Why are they still eigenvectors? (Assuming that the matrix in question is normal)
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Finding the basis that results from an inner space

I'm given the inner product: $\bigl\langle(x_1,x_2,x_3),(y_1,y_2,y_3)\bigr\rangle:=3x_1y_1+x_1y_3+y_1x_3+x_2y_2+2x_3y_3$ And I'm asked to find the orthonormal basis in respect of the above inner ...
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Polynomial Approximation with Odd an Even Functions

Remember that a function $f$ is called even if $f(−x) = f(x)$ and odd if $f(−x) = −f(x)$ for all $x$ in its domain. Let $w$ be an even weight function on the interval $(−a, a)$ and ${ϕ_0, ϕ_1, .., ϕ_n}...
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Gram Schmidt process on a polynomial

I'm currently learning Linear Algebra and I was asked to calculate the orthogonal projection of the vector $x^3$ on a subspace of $R_5[x]$ - $U = Sp\{1, x, x^2\}$ with the integral dot product in $[0,...
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Every separable Hilbert space has an orthonormal basis

Prove the following: Every non trivial separable Hilbert space $H$ has an orthonormal basis, i.e., an orthonormal set whose linear span is dense in $H$ My attempt: Let $V = (v_n)_{n\geq1}$ be a ...
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Determine Orthonormal basis in $R^4$ without calculator

In $R^4$ with the standard inner product, the linear subspace $U$ is given by: $$U = <(0,1,2,2), (2,1,0,-1), (0,0,0,3)>$$ The linear subspace $V$ of $U$ consists of those elements in $U$ ...
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Dimension of an orthonormal basis

I'm currently completing an assignment for my linear algebra class, and I feel that I don't completely understand the dimension of an orthonormal basis. For reference, this is the question I am ...
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History Question - QR Factorization

I'm about to teach QR factorization and I would be curious to know why the orthogonal matrix is typically denoted $Q$ and the upper triangular matrix is denoted $R$
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Orthonormal Basis of Hyperplane

Find an orthonormal basis of the hyperplane $x_1 + x_2 + x_3 + x_4 + x_5 = 0$. So I understand how to use Gram-Schmidt to solve this, but I'm having issues finding the basis to start with. Is it as ...
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334 views

Modified Gram Schmidt

How does the Modified Gram Schmidt works? I want to use it but I am confused by the notations and I could not find any example online.
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Gram-Schmidt orthonormal basis

Given: I am given four different signals \begin{align} s_0(t) = \begin{cases} 2, \ \ \ \ 0 < t \leq 1 \\ -2, \ 1 < t \leq 2 \\ 2, \ \ \ \ 2 < t \leq 3 \\ \end{...
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1answer
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Gram Schmidt Process with inner product $\langle z,w\rangle = 3(z_1)(\bar{w_1}) + 2(z_2)(\bar{w_2})+i(z_1)(\bar{w_2})-i(z_2)(\bar{w_1})$

We are given an example where we have $v_1 = (1,0)$ and $v_2=(0,1)$ on the complex plane with the inner product $\langle z,w\rangle = 3(z_1)(\bar{w_1}) + 2(z_2)(\bar{w_2})+i(z_1)(\bar{w_2})-i(z_2)(\...