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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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 ...
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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 :
$$...
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Distance between point and hyperplane given basis
I used the Gram-Schmidt process to find the orthonormal basis for some hyperplane, $V$, in $\mathbb{R}^4$. The vectors are
$$
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
$$S' = \{1/||1||, x/||x||, x^2/||x^2||\}$$
...
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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 ...
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