# Questions tagged [orthonormal]

For questions related to orthonormality. A set of vectors in an inner product space is called orthonormal if each vector is a unit vector, and vectors are pairwise orthogonal.

799 questions
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### Equivalence of statements about bounded linear maps on a Hilbert space

Assume $H$ is a Hilbert space and $V \in L_b(H)$. I want to show, that the following propositions are equivalent: V is an isometry. For every orthonormal system $\{u_{\alpha}: \alpha \in A \}$ the ...
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### Can the eigenvectors of a linear operator in an infinite-dimensional space span the space and be linearly dependent at the same time?

Consider a vector space $V$ over the complex field which is infinite-dimensional with a Euclidean inner-product. Let $L$ be a linear operator on $V$. Say a subset of eigenvectors of $L$ forms a ...
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### Find which conditions must parameters $a$ and $b$ meet so there's exist an orthonormal basis

In $\mathbb{E^3}$ we have the plane $\pi:x-y+z-3=0$, the line $r:(2,0,1)+t(1,1,0),\ t\in\mathbb{R}$, and the point $P=(3,0,3)$. Which conditions must parameters $a$ and $b$ meet so there's exist an ...
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### Inner product that makes vectors an orthonormal basis

Let $X= \begin{pmatrix} a \\ b \end{pmatrix}$ and $Y=\begin{pmatrix} c \\ d \end{pmatrix}$ be two vectors in the plane. Do we have the existence of an inner product that makes ...
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### Write $(-1,1,2,2)^T$ in terms of the basis $U$

I have found that $U=\{1/2\pmatrix{1\\1\\1\\1},1/2\pmatrix{-1\\-1\\1\\1},\frac{1}{\sqrt{2}}\pmatrix{-1\\1\\0\\0},\frac{1}{\sqrt{2}}\pmatrix{0\\0\\1\\-1}\}$ is an orthogonal basis for $\mathbb{R}^4$. ...
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### Distribution of column of random orthogonal matrix

Suppose $O \in \mathbb{R}^{n \times r}$ with $r < n$ is a random matrix whose distribution is uniform on the set of $r \times n$ matrices such that $O'O = I_r$. Is is true that the columns of $O$...
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