# 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 ...
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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 ...
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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.
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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 ...
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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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### 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?
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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 $\lim_{n\to\infty} (\vec{... 0 votes 1 answer 104 views ### Building orthogonal vectors Suppose I have$N$vectors$v_1, v_2, v_3, ..., v_N$I already know that I can build an orthonormal basis with the Gram-Schmidt process, but I need to keep the first three fixed, even if they are not ... 4 votes 1 answer 72 views ### Can vectors be rearranged such that all Gram-Schmidt coefficients have absolute value$\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$be the orthogonal vectors obtained from the Gram-Schmidt process (without ... • 810 0 votes 3 answers 114 views ### Why are the angles between columns of$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$. $$G = A^\top A = (QR)^\top(QR) = R^\top Q^\top QR = R^\top (Q^\top ... • 1 4 votes 1 answer 122 views ### Gram Schmidt process is a homotopy equivalence How can one see that the Gram Schmidt process is a homotopy equivalence SL(m,\mathbb{R})\rightarrow SO(m)? Intuitively it seems plausibel to me, but I can't find an explicit homotopy between the ... • 366 0 votes 0 answers 44 views ### Use Gram-Schmidt to find an orthonormal basis of a product of two polynomials. Given n = 4, x_1 = 1, x_2 = 0, x_3 = 2, x_4 = 3, and B = span\{1, t, t^2\}, find an orthonormal basis of B under the scalar product \sum_{i=1} ^{n} p(x_i)q(x_i) using Gram Schmidt. I am very new ... • 67 1 vote 1 answer 37 views ### The Gram-Schmidt process: on the equality of the generated subspace. We work in an infinite Hilbert space H. Let (x_k) an arbitrary linearly independent sequence, then we define$$e_1=\frac{x_1}{\lVert x_1\Vert}\quad\text{and}\quad e_k=\frac{x_k-\sum_{j<k}\... • 150 1 vote 0 answers 52 views ### 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\|...
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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 ...