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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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Gram-Schmidt process: kernel and image

Let $(V, \langle . , . \rangle)$ be a euclidian vector space and let $w \in V$ with $\Vert w \Vert = 1$. Let $f$ be a map with $f: V \to V, v \mapsto v - \langle v, w \rangle w$. I'm supposed to find ...
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Finding inner product associated with an orthogonal basis

Let $S = \{1 + k_0, c_1x + k_1, c_2x^2 + k_2, c_3x(x^2 - 3) + k_3\}$ ($x(x^2 - 3)$ is so that its derivative is proportional to $(1-x)(1+x)$), $-1 \leq x \leq 1$. Is it possible to find an inner ...
LaguerreGroup's user avatar
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After applying Gram Schmidt to orthonormalize columns that are random normal gaussian vectors what is the resulting distribution?

From reading through responses in these answers here and here, I know that if I have a gaussian random vector $X \sim \mbox{Norm}(0,1)$ that $X^2$ should have a Chi squared distribution. I also think ...
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Basis of $V$ that contains a certain element $v$ [closed]

I'm a student learning linear algebra. Few days ago, I studied the concept of Gram-Schmit procedure in a given inner product space. With this procedure, if any $v \in V$ is given, then we can ...
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Orthonormal Basis of subspaces of inner product spaces

I want to show the following: Let W be a subspace of the inner product space V. Then W has an orthonormal-Basis that is subset of a orthonormal-Basis of V. So if V was finite the statement could be ...
Dave's user avatar
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Extend an orthogonal set of vectors to an orthonormal basis in SVD.

I'm learning about Singular Value Decomposition (SVD) and how to compute each matrix in the decomposition $A=U\Sigma V^T$. I know how to compute $\Sigma$ and $V$ and hence most of $U$, since we know ...
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Is the Gram-Schmidt Orthogonalization process for functions the same as it is for vectors? [closed]

I was not able to find resources online and was wondering so since it would greatly help me with my work if I can directly apply what I have learned from linear algebra.
azozer's user avatar
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Are Orthogonal Basis perpendicular to Original Basis in Gram-Schmidt?

a, b and c are 3 Independent Vectors. We can generate Orthonormal basis vectors using those 3 vectors using Gram-Schmidt method. Lets say those 3 orthogonal basis vectors generated from a, b and c are ...
Abhishek's user avatar
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If $(v_n)_n$ is a Hilbert basis for $H$ over $\mathbb{C}$, can you turn $(\mathrm{Re}(v_n), \mathrm{Im}(v_n))_n$ also to a Hilbert basis for $H$?

(Question:) Suppose that $H$ is a Hilbert space over the field $\mathbb{C}$ and that $(v_n)_{n=1}^\infty$ is a Hilbert basis for it, that is a sequence of orthonormal vectors whose span is dense in $H$...
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Write vector as the sum of w and w orthogonal [closed]

Let $\mathbb{R}^3$ be given with the standard inner product and let $W$ be the subspace spanned by $\left( \begin{array}{c} 4\\ -2\\ 4\\ \end{array} \right)$ and $\left( \begin{array}{c} -2\\ 6\\ 2\\ \...
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How to express a Gaussian as a series of exponential? $\displaystyle e^{-x^2}=\sum_{n=1}^{\infty}c_n e^{-nx}$

Context I would like to express the Gaussian function as a series of exponentials: $$e^{-x^2}=\sum_{n=1}^{\infty}c_ne^{-n|x|}\qquad\forall x\in\mathbb{R}$$ For simplicity (the absolute value is added ...
Math Attack's user avatar
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What do the parts of the Gram-Schmidt process mean and represent in space?

I am struggling to understand what the different parts of the Gram-Schmidt process represent. Suppose we have a basis $\{x_1, x_2\}$ We would then find a orthogonal basis by doing the following : $$...
Yassine's user avatar
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Distance between point and hyperplane given basis

I used the Gram-Schmidt process to find the orthonormal basis for some hyperplane, $V$, in $\mathbb{R}^4$. The vectors are $$ u = \begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix}, v = \begin{bmatrix}1\\ 1\\ ...
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Gram-schmidt process for polynomials [closed]

$$S = \{1, x , x^2\}$$ S is a set of orthogonal vectors. So instead of applying the gram-schmidt process to obtain the orthonormal basis, can't we just do $$S' = \{1/||1||, x/||x||, x^2/||x^2||\}$$ ...
Samyak Jain's user avatar
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Question about Theorem 1.13 in Tom Apostol's Calculus Vol 2.

I'm almost sure that I'm misreading the theorem, but I have no idea where. Theorem 1.13 is about orthogonal bases and Gram-Schmidt process. It states: Let $x_1, x_2, ...., $ be finite or infinite ...
Mr. Proper's user avatar
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Cholesky factorization using Gram-Schmidt

I am trying to find the Cholesky factorization $AA^T$ of the below covariance matrix $C$, to decompose a gaussian vector into independent standard normal random variables. However, the entry $a_{(3,3)}...
Quasar's user avatar
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$x \cdot F_n$ is a linear combination of $\{ F_{n−1}, F_n, F_{n+1} \}$ for every positive integer $n$.

Let $F_n$ denote the $n$th polynomial obtained through Gram-Schmidt orthogonalization applied to the sequence $$\{ 1, x, x^2, x^3 , \cdots , x^n \}$$ with the inner product $$ \int_0^1 f(x) \cdot g(x) ...
An Isomorphic Teen's user avatar
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Answer Check: Best Approximation of $(x+1)e^{-x}$ using Legendre Polynomials

I have worked through and produced an answer for the following question but am unsure of whether or not it is correct. I would appreciate some insight. $\textbf{Question}$: Using the Gram-Schmidt ...
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Got stuck on Finding an orthonormal basis for the orthogonal complement

The problem is Find an orthonormal{w_1, w_2} for the orthogonal complement of the subspace U given by 2x+y-2z=0 2y-x+2z+w=0 I can not reach out next step from here.... enter image description here
Burce's user avatar
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Finding mutually orthogonal vectors in a given set

First I'd like to give you the context of my question (you can skip this paragraph if you like). In this paper on SIC-POVM, the authors show sets of fiducial vectors (section IV.) used to explicitly ...
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Constructing rotation matrix with respect to central point

I am trying to implement an algorithm which has three points $z_0, z_1, z_2$ and creates a rotation matrix with respect to $z_0$. According to the paper, "we may apply a Gram-Schmidt process to ...
anirudhvca's user avatar
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Is this a simpler way to orthogonalize bases than Gram-Schmidt?

This is my first post here. I have a question. In my linear algebra course we are learning the Gram-Schmidt process. But it appears to me in a much more intuitive way to do the cross product ...
From ARGENTINA's user avatar
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show that for every independent vectors $u,v\neq0$ there exist w such that $\left\langle u,w\right\rangle <0,\left\langle v,w\right\rangle >0$

I need to show that for every independent vectors $0\neq u,v\in\mathbb{R}^{n}$ there exist $w\in\mathbb{R}^{n}$ such that $\left\langle u,w\right\rangle <0,\left\langle v,w\right\rangle >0$ I ...
Liel Azulay's user avatar
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1 answer
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Using Gram-Schmidt on independent vectors to find a special vector $w$

Let $u,v \in \mathbb{R}^{n}$ be non-zero independent vectors. Show that there exists a vector $w \in \mathbb{R}^{n}$ such that $\langle u,w \rangle < 0$ and $\langle v,w \rangle > 0$. I was ...
Tom Waller's user avatar
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Find the basis the transformation T is for given its actions?

I need help with b, but I also need to confirm something about a. Lets talk about b, lets call the column vectors of the transformation matrix $w_1, w_2, w_3$. I can already see that $w_3 = [1, 2, 2]^...
Need_MathHelp's user avatar
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Orthonormal basis in a space with an isotropic inner product

Let's $V$ be a space with a generic inner product $\langle \,\cdot \mid \cdot\, \rangle$. Let's extend the concept of orthonormal base: a base $\mathcal{B}$ for the vector space $V$ is orthonormal iff ...
Michele Cantarelli's user avatar
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Orthogonal bases and Gram–Schmidt

What is the closest function $a \cos(x) + b \sin(x)$ to the function $ f(x) = \sin(2x)$ on the interval from $-\pi$ to $\pi$? What is the closest straight line $c + dx$? This is a problem in linear ...
Vinay Mahesh's user avatar
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104 views

Best approximation in $L_2$

I can't find the error. $$H=L_2[-1,1], \; w(t)=t^2 ~- \text{weight function}, \; x_1(t)=1, \; x_2(t)=t^2,\; x_3(t)=t^4, \\ x_0=\cos(t).$$ It is necessary to find the best approximation for $x_0$. I ...
frumple's user avatar
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Completing an orthonormal collection to an orthonormal basis in $\mathbb{R}^n$

Let $\left\{q_1, q_2, \dots, q_k\right\} \subset \mathbb{R}^n$, $k<n$, be an orthonormal collection. Let $\left\{e_1, e_2, \dots, e_n\right\} \subset \mathbb{R}^n$ be the standard (orthonormal) ...
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How to generate an adapted frame for arbitrary curve?

I was reading a paper about Bishop frame and try to calculate the Bishop frame for a cubic bezier curve. My plan to do so is to generate an arbitrary adapted frame $\left\{T, N_1, N_2 \right\}$ and ...
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Gram–Schmidt process: matrix form

Let $$ \operatorname{proj}_{\mathbf{u}}(\mathbf{v})=\frac{\langle\mathbf{u}, \mathbf{v}\rangle}{\langle\mathbf{u}, \mathbf{u}\rangle} \mathbf{u} $$ be the projection operator, where $\langle\mathbf{u},...
Mark's user avatar
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1 answer
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Find a basis of $\mathbb{R}^2$ that is orthogonal for $q_1$ and in which $q_2$ is diagonal.

On $\mathbb{R}^2$, with coordinates $(x, y)$ consider the quadratic forms $q_1$ and $q_2$ defined by $q_1(x, y, z)=2 x^2+4 x y+3 y^2+2 y z+2 z^2$ and $q_2(x, y, z)=x y+x z+z^2$. Find a basis of $\...
vitalmath's user avatar
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Multiple regression by successive orthogonalization

I was studying The Elements of Statistical Learning book and trying to understand the section where multiple linear regression is explained by successive orthogonalization procedure, i.e. Gram-Schmidt ...
ConventionalProgrammer's user avatar
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Dimensions of non square matrix?

After performing Gram-Schmidt process for the following matrix: $\pmatrix{3&2&3\\ 2&5&-1\\ 2&4&8\\ 12&2&1}$ the resultant matrix is this one: $\pmatrix{0.23643312&0....
Lerner Zhang's user avatar
1 vote
1 answer
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Completing a set of orthonormal vectors in $\mathbb{R}^n$

Suppose that I have an orthonormal set of vectors $$ \left\{{\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \right\}  \subset \mathbb{R}^n, $$ where $k < n$. How can I use Gram-Schmidt orthonormalisation ...
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Intuitive explanation of dividing outer product matrix by inner product

For some vector $v$, the matrix $P=vv^T$ is an orthogonal projector that projects onto the line on which $v$ lies. What is the intuitive meaning of the matrix $M$ obtained by dividing $P$ by $v^Tv$? $$...
Ahmir_r4's user avatar
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Proof of loss of orthogonality in Gram-Schmidt

I am stuck at understanding about how to derive the following proofs related to error bounds which are given in the following slides. Can anyone please explain to me how these are derived?
ThickThighs's user avatar
1 vote
3 answers
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Find an orthogonal matrix $Q$ and a diagonal matrix $D$ such that $A=QDQ^T$

Let $$A=\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}$$ Find an orthogonal matrix $Q$ and a diagonal matrix $D$ such that $A=QDQ^T$. I already got these Eigenvalues $D=\...
Donald's user avatar
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Arnoldi decomposition/ algorithm: Uniqueness

The Arnoldi decomposition states the following: let $A \in \mathbb{C}^{n×n}$ and $v ∈ \mathbb{C}^{n} \setminus \{0\}$ be of grade $d$ with respect to $A$. Then there exists $V \in \mathbb{C}^{n×d}$ ...
Pazu's user avatar
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Finding and visualising orthonormal basis

I am solving this problem 19 ( Sorry for uploading picture i ain't that good with latex. please help me out ! ) Here rank of Matrix is two so answer will be (b) or (d) . I am trying to visualise this ...
RKK's user avatar
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Signature and Orthogonal Basis of Symmetric Bilinear Form

I am trying to find the signature and a orthogonal basis for the symmetric bilinear form $(x,y) \mapsto x^tAy$ where $A$ is $\begin{bmatrix} 5 & 5 & 0 \\ 5&-8&-2 \\ 0&-2&0 \end{...
Aris Konstantinidis's user avatar
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0 answers
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Prove (or disprove) that limit is preserved under Gram-Schmidt process

I am trying to prove or disprove the following statement: For a general inner product space $(V, \langle\cdot,\cdot\rangle)$, if a sequence $(\vec{w}_n)_n$ in $V$ is such that $\lim_{n\to\infty} (\vec{...
Paatryk.Y's user avatar
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1 answer
104 views

Building orthogonal vectors

Suppose I have $N$ vectors $v_1, v_2, v_3, ..., v_N$ I already know that I can build an orthonormal basis with the Gram-Schmidt process, but I need to keep the first three fixed, even if they are not ...
Enrico Detoma's user avatar
4 votes
1 answer
72 views

Can vectors be rearranged such that all Gram-Schmidt coefficients have absolute value $\le 1$?

Consider a set of linearly independent vectors $v_1, \cdots, v_p \in \mathbb{R}^n$. Let $u_1, \cdots, u_p \in \mathbb{R}^n$ be the orthogonal vectors obtained from the Gram-Schmidt process (without ...
Brian Lai's user avatar
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3 answers
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Why are the angles between columns of $A$ the same as angles between the columns of $R$ when $A=QR$ and $A^\top A=R^\top R$?

Suppose a matrix $A$ has linearly independent columns and QR factorization $A = QR$. I was able to prove that $A^\top A = R^\top R$. $$G = A^\top A = (QR)^\top(QR) = R^\top Q^\top QR = R^\top (Q^\top ...
Hans's user avatar
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4 votes
1 answer
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Gram Schmidt process is a homotopy equivalence

How can one see that the Gram Schmidt process is a homotopy equivalence $ SL(m,\mathbb{R})\rightarrow SO(m)$? Intuitively it seems plausibel to me, but I can't find an explicit homotopy between the ...
Léo Mousseau's user avatar
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Use Gram-Schmidt to find an orthonormal basis of a product of two polynomials.

Given $n = 4, x_1 = 1, x_2 = 0, x_3 = 2, x_4 = 3, and B = span\{1, t, t^2\}$, find an orthonormal basis of B under the scalar product $\sum_{i=1} ^{n} p(x_i)q(x_i)$ using Gram Schmidt. I am very new ...
No_Bass's user avatar
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1 answer
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The Gram-Schmidt process: on the equality of the generated subspace.

We work in an infinite Hilbert space $H$. Let $(x_k)$ an arbitrary linearly independent sequence, then we define $$e_1=\frac{x_1}{\lVert x_1\Vert}\quad\text{and}\quad e_k=\frac{x_k-\sum_{j<k}\...
NatMath's user avatar
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Modified Gram-Schmidt

The algorithm for modified Gram-Schmidt (it does the same thing, but is supposed to be better for computers due to rounding) is: For k=1,...,m set $\tilde x_k=k_k$ Set $\tilde x_1=\tilde x_1/\|x_1\|...
Vons's user avatar
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1 vote
1 answer
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Is Gram-Schmidt process redudant about square full-rank matrices?

I am trying to grasp the concept of Gram-Schmidt process and I have encountered the following logical difficulty: Given a set of $n$ independent vectors, applying GS algorithm upon this set would ...
Ohad's user avatar
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