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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Gram-Schmidt orthonormal basis for x1 + x2+ x3 =0

I need to find an orthonormal basis for the plane x1 + x2 + x3 = 0 by using Gram-Schmidt on the vectors $$ \begin{pmatrix} 1 \\-1 \\0 \end{pmatrix}, \begin{pmatrix} 0 \\1 \\ -1 \end{pmatrix},\begin{...
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Finding Orthonormal Basis using SVD and comparing it with Gram-Schmidt shows different result

I was trying to find the orthonormal basis for the column space of the following matrix "A" \begin{pmatrix} -1 & -1 & 2 & 3 \\ -1 & 1 & -3 & -4 \\ 2 & -2 & 5 ...
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Recursive formula on Gram-Schmidt polynomials

Construct the sequence $\{p_n\}_{n\in \mathbb{N}_0}$ by applying the Gram-Schmidt procedure on $\{x^n\}_{n\in \mathbb{N}_0} \in L^2([0,1])$ whitout normalizing so that they are monic ($p_0=1$ , $p_1=x-...
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Why calculate the projection of V_k in Gram-Schmidt when its length and angle are fixed?

I suspect this is a very naive question. When projecting the next subspace onto the previous in the Gram-Schmidt process, since it is already known it has to be normalized to length 1, has to be ...
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51 views

Find all polynomials in a vector space orthogonal to another polynomial with Gram Schmidt possibly

Here I am given the following. Define an inner product in $\mathcal{P}_2$ by: $$\left<p,q\right>=p(-1)q(-1)+p(0)q(0)+p(1)q(1).$$ a) Starting with the basis $\left\{1,x,x^2\right\}$ of $\mathcal{...
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32 views

Gram-Schmidt orthogonalization: finding an orthogonal set

Does anyone know how to solve this problem? Let $f_{1}(x)=x, f_{2}(x)=x^{2}, f_{3}(x)=x^{3} \in L^{2}[0,1]$ , Find an orthonormal set from $\left\{f_{1}(x), f_{2}(x), f_{3}(x)\right\}$ by using the ...
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69 views

Using Gram-Schmidt to find an orthonormal basis.

I am studying Linear Algebra and have been introduced to Gram-Schimdt algorithm and I am struggling on how to approach this question. For the beginning inner product I'm unsure how you get the answer:...
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Gram-Schmidt orthogonalization procedure - prove some properties

Let V be vector space in Rˆn. Using Gram-Schmidt's orthogonalization procedure notations, show that: a) if j >= 2, v_j is perpendicular to v_i for every i < j and v_j != 0. b) the set C = { v_1, ...
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35 views

Compute iterated kernels for symmetric kernel $K(x,t)=\sum_{n=1}^\infty \frac{\sin(n\pi x) \sin(n \pi t)}{n}.$

Compute iterated kernls for symmetric kernel $$K(x,t)=\sum_{n=1}^\infty \frac{\sin(n\pi x) \sin(n \pi t)}{n}.$$ I am studying integral equations with symmetric kernels and there is a given problem. ...
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Finding the orthonormal basis of $\Bbb R^3$ using the Gram-Schmidt algorithm

I'm relatively new to inner product and I was asked to obtain an orthonormal basis of $\Bbb R^3$ using $$ \begin{pmatrix} 1 \\ 1 \\ 1 \...
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287 views

Intuitive explanation of why the modified Gram-Schmidt is more stable than the classical one?

This may be an old question, and there are certainly some related posts which I will mention below. However, there seems no clear answer to me yet. The question is: is there an intuitive way to ...
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57 views

Gram-Schmidt process for polynomial space

I'm looking for a orthonormal basis to 2 polynomials $f,g ∈ \Bbb R_2[x]$ we will define inner product space: $<f,g>=f\left(0\right)\cdot g\left(0\right)+f\left(1\right)\cdot \:g\left(1\right)+f\...
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69 views

Gram-Schmidt Process: How does it fail?

I am confused about the Gram-Schmidt process for a linearly dependent set. I can't seem to grasp why it would cause the linearly dependent vectors to equal zero. Can someone please explain?
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Problem with orthogonalizing the Laguerre polynomials

Alright, so I ran into a little problem while applying the Gram-Schmidt orthogonalization process. To the functions $\{1,x,x^2,x^3...\}$ over $x\in(0,\infty)$ with weight function $\sigma (x)=e^{-x}$. ...
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Linear Algebra Gram Schmidt

A plane $\pi$ in $R^3$ is spanned by $[2, 4, 4]^T$ and $[5,4,-2]^T$ Find the linear transformation that obtains the closest point on the plane from any point in $R^3$. Find the volume of tetrahedron ...
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Geometric Motivation for Inner Product

I think some background will make the kind of answer I'm looking for clearer. I'm trying to think of an elementary proof of the Pythagorean Theorem. I don't like the geometric proofs because they all ...
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1answer
36 views

Gram Schmidt process for defined polynomials

I would like to know whether I have made mistakes in this Gram-Schmidt process as I kept getting mixed up with the vectors. Using the vector space $P_1$ defined by the inner product $<p, q> = ∫_{...
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Trouble with the Gram-Schmidt Orthogonalization Derivation

In the book "Linear Algebra as an Introduction to Abstract Mathematics", Pages 102-103 outline the constructive proof used to illustrate the feasibility of generating an orthonormal list of ...
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Proving a step in gram schmidt process in induction

I have the following matrix $n\times n$: $$ A=\begin{bmatrix} 1 & 1 & 1 & 1 & \ldots & 1\\ 0 & 1 & 1 & 1 & \ldots & 1\\ 0 & 0 & 1 & 1 &...
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Gram-Schmidt method to get a basis for $P_3$

If $P_3$ is a vector space of third-degree polynomials. It is known the basis for $P_3$ is ${( 1,x,x^2 , x^3})$ and $\langle p, q\rangle = \int_{0}^{1} p(x)q(x)\, dx.$ is a valid product on $P_3$ I ...
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Can it be possible to get a mutually orthogonal vector in this case?

Suppose $\mathbf{v}_1$ and $\mathbf{v}_2$ are real vectors of length $N>3$. If we use Gram-Schmidt process, we can find two orthogonal vectors $\mathbf{u}_1, \mathbf{u}_2$ such that $$ \mathbf{u}_1 ...
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Using Gram-Schmidt to find an orthogonal basis

I was given the following question: The given set is a basis for subset $W$. Use the Gram-Schmidt process to produce an orthogonal basis for $W$.$$\left\{\left[\begin{matrix}0\\8\\8\\\end{matrix}\...
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How to derive $u^Tu\cdot v-v^Tu\cdot u$

In this answer, the author states that $u_\bot = u^Tu\cdot v-v^Tu\cdot u$ is perpendicular to $u$ if $v$ not parallel to $u$. This seems derived from the Gram–Schmidt process, and I can indeed derive ...
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26 views

Orthogonalizing a basis into the standard basis of $\Bbb{R^n}$

The problem is to orthogonalize the following basis in $\Bbb{R^n}$. $$e'_1=(1,0,...,0)$$$$e'_2=(1,1,...,0$$ $$\cdots$$ $$e'_n=(1,1,...,1).$$ Lets denote the orthogonal basis elements with $e_i$. Let $...
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59 views

Show that if the Gram–Schmidt process is applied to a linearly dependent vector system, then it outputs the 0 vector

If the Gram–Schmidt process is applied to a linearly dependent vector system, it outputs the 0 vector Can somebody help me with this problem I thought taking the linear combination of the linearly ...
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23 views

Projection of a function onto the orthogonal complement of a subspace

I have a polynomial subspace, lets say $U$, which I applied the Gram-Schmidt algorithm to find an orthonormal basis. I had to find the projection of $cosh(x)$ onto $U$, so as I had found the ...
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42 views

Gram-Schmidt procedure for new base, complete positivity

Let say that $v_1,v_2,...v_n$ be vectors in a $m$-dimensional Euclidian space $V$. There exists a natural number $k$ and an isometry $T:V \Rightarrow R^k$ such that $$Tv_1,Tv_2,..., Tv_n \in R^k_+=\{...
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Linear Algebra: Gram-Schmidt $A=QR$ factorisation and diagonal components of $R$

With the $A=QR$ factorisation ($Q$ is a matrix with orthonormal columns $q_1$ to $q_n$, $A$ is a matrix with independent columns $a_1$ to $a_n$. $Q$ is created from the columns of $A$ with Gram-...
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transform the basis vectors into an orthonormal basis.

Let 𝑅3 have the Euclidean inner product. Use Gram-Schmidt process to transform the basis vectors 𝑢1=(1,0,0),𝑢2=(3,7,−2),𝑢3=(0,4,1) into an orthonormal basis I was able to find alpha1=(1/√2,0,0) ...
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Omit an element of a Gram–Schmidt process

Let $H$ be an Hibert space over $\mathbb{C}$ Let $\{h_n\}_{n \in \mathbb{N}} \subset H$ a sequence of linearly independent vectors in $H$ such that $h_n \to h$ in norm topology. We apply Gram–Schmidt ...
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How does Gram–Schmidt process produce normalized, orthogonal rows?

Assuming a square matrix, I see how this process would produce orthonormal columns, but I cannot see the rows also turn out to be orthonormal after this process? It seems like some kind of witchery. ...
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Confused regarding the Gram-Schmidt Theorem.

I'm given the scalar product in $\mathbb{R}^3$ defined as $$\langle (x_1,x_2,x_3), (x_1,x_2,x_3)\rangle=x_1^2+2x_2x_1+2x_2^2+x_3^2$$ Let $B=\{ v_1,v_2,v_3\}$ a basis of $\mathbb{R}^3$ where $v_1=(-1,1,...
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Gram-Schmidt orthogonalization. Stuck with an example, can someone show me where I am going wrong?

I am pretty sure I am missing something somewhere but I cannot see where and have been stuck for a while... Like in the example I set the first vector $U_1$ to \begin{pmatrix} -1\\ 1\\ 0\\ ...
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Ortogonal basis for a subspace of $R^{4}$ and find a point in a plane closest to the origin.

Let $V \subset R^{4}$ a subspace defined for the equation $x_{1}+3x_{2}-5x_{3}-x_{4}=0$. a) Find a ortogonal basis for $V$. b) Wich point in the plane $x_{1}+3x_{2}-5x_{3}-x_{4}=36$ is closest to $(...
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Gram Schmidt process

We have previously calculated the basis to be: $\left( \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\right)$. We ...
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Unitary matrix U to diagonalize matrix A

I'm working on this exercise and I got stuck. Find a unitary matrix $U$ and a diagonal matrix $D$ such that $A=U^{*}DU$ for $$A=\begin{pmatrix}1&i\\-i&1\end{pmatrix}$$ So what I decided to ...
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Why doesn't Gram-Schmidt work for all matrices?

So I'm doing a MATLAB assignment where the objective is to write code to do the Gram Schmidt algorithm. I just want to say, my code is almost certainly not the issue, as the same issue arises in ...
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Orthonormalization of basis - Diagonalization of overlap matrix.

Is it sufficient to diagonalize overlap matrix (Gram matrix) in order to find orthogonalized basis? Overlap matrix $G$ can be diagonalized as $P^{-1} G P=\lambda$. My new set of vectors would be $\...
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Explicit formula for orthonormalized vectors (using Gram-Schmidt)

For $n\in\mathbb{N}$, let $$ B(n):=\{b_1,b_2,\ldots,b_n\}:=\left\{\begin{pmatrix}1\\2\\3\\4\\\vdots\\n\end{pmatrix},\begin{pmatrix}2\\-1\\0\\0\\\vdots\\0\end{pmatrix},\begin{pmatrix}3\\0\\-1\\0\\\...
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Projection onto a subspace in an $L_p$ normed space

I am trying to write down the angle between two $1$-dimensional and $t$-dimensional subspaces of a normed space $L_p$. In particular, I am following Milicic's On the Gram-Schmidt Projection in Normed ...
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Can Gram-Schmidt algorithm be applied to every set of vectors in ${\rm I\!R}^{n}$?

I have the following question: "Can Gram-Schmidt algorithm be applied to every set of vectors in ${\rm I\!R}^{n}$?".I know that in the general case we apply it to the finite independent set of the ...
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Use the Gram-Schmidt process to find the orthonormal basis for the row space of the matrix $A$.

Use the Gram-Schmidt process to find the orthonormal basis for the row space of the matrix $A$. The matrix $A$ is as follows: \begin{bmatrix}1&1&0&0\\-1&3&0&1\\-3&1&-...
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Find the orthonormal basis for the subspace $U$ of $M_{2,2}(\mathbb{R})$ spanned by

Consider the real inner product space $M_{2,2}(\mathbb{R})$ (the space of 2 x 2 matrices with real entries), with inner product: (a) Find the orthonormal basis for the subspace $U$ of $M_{2,2}(\...
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Prove the Gram-Schmidt process for an inner product space.

Consider set of arbitrary non-zero vectors $\{v_1,...,v_n\}$ in an inner product space $V$. Consider set $S = \{p_1,...,p_n\}$, where $p_1 = v_1$ and $$p_j = v_j - \biggr[\sum_{x=1}^{j-1}\frac{\...
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Gram-Schmidt algorithm and degenerate bilinear forms

I am studying Gram-Schmidt algorithm on Euclidean spaces(real spaces with a positive definite bilinear form). And I was wondering if this algorithm was valid also in more general spaces. I think that ...
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Intuition for Gram-Schmidt process

Let V be a finite dinner-product space, given a$ v_1,...v_n$ orthogonal bases, and $w_1,...w_n \in W $be subspace of V. Then $v_1=w_1$, $v_2= w_2-proj_w v_2$, $v_3= w_3-proj_w v_2- proj_w v_3$... I ...
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Is the Gram-Schmidt procedure a bijection?

Since the Gram-Schmidt procedure yields a single orthonormal output for a given linearly independent input, we can view the procedure as a function which maps an original linearly independent list of ...
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69 views

Gram-Schimidt process(trying to understand the projection operator)

For Gram-Schmidt process, the projection operator is defined by $proj_u(v)=\frac{<v,u>}{<u,u>}u$. Can someone give me an reference for interpreting this projection operator? What do $<v,...
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Uniqueness in Gram-Schmidt process

Until now, I had only used the Gram-Schmidt process to prove the existence of an orthonormal basis. However, I stumbled upon an exercise which uses the full theorem, that is to say : Let $E$ be a pre-...
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Prove that $[T]_\gamma$ is an upper triangular matrix (Question 6.4.24 (a) in Linear algebra by Friedberg, Insel, and Spence).

Question 6.4.24 (a) (in Linear algebra by Friedberg, Insel, and Spence): Suppose that $\beta$ is an ordered basis for $V$ such that $[T]_{\beta}$ is an upper triangular matrix. Let $\gamma$ be the ...