# Questions tagged [orthogonality]

This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.

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### Is there any relation between these polynomials and a set of orthogonal polynomials?

as you know for every set of orthogonal polynomials as $P_0(x)=1, P_1(x), P_2(x),..., P_n(x), ...$ we have exactly $n$ real roots for $P_n(x)$ and also the fact that $\int_a^b P_i(x)P_j(x)d\alpha(x)=0$...
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### Is it true that $T$ is orthogonal if and only if $T$ is isomorphism?

I want to prove the following: Let $V$ be a finite dimensional vector space with inner product, then $T$ is orthogonal if and only $T$ is an isomorphism I think the sufficiency could be true because ...
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### Why does $T(x)=2x$ keep angles but is not an isometry? [closed]

I would like to know why a defined operator $T$ on $V$, as following: $T(\alpha)=2\alpha$ preserves angles but is not an isometry?
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### How to calculate angle in non-orthogonal system (B angle at 45 degree, C angle attached on B) based on orthogonal angles

I am trying to find a formula to recalculate rotation from orthogonal BC system to defined non-orthogonal BC system In which B is shifted by 45 degree. Graphic example below: non-orthogonal system B ...
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### Orthogonal tranformation over invariant subspace

let $V$ be a euclidean space. $T:V\to V$ be an orthogonal linear transformation. $W\subset V$ is a $T$-invariant subspace. I need to prove 2 things: A. $T\bigl|_W$ is orthogonal so I said that if ...
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### Finding orthogonal projection on Hilbert space

Let $H = L^2(−1, 1)$ and $L \subset H$ be the set of all continuous functions such that $f(0) = 0$. Find the orthogonal projection $P : H → \bar{L}$. My thoughts on this: We say $g=P_{\bar{L}}(f)$ ...
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### Range of a unitary transformed orthogonal projection

Let $X$ be a Hilbert space, $P$ an orthogonal projection in $X$ and $Q \in L(X)$ (i.e. $Q \colon X \to X$ is linear and continuous) a unitary linear transformation, i.e. $Q^*Q=Id_X= QQ^*$ ($Q^*$ ...
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### Rank of a left-orthogonal decomposition $A^T=UB^T$

I've got a rather simple question, but couldn't find an answer to it: Say $A\in\mathbb R^{m\times n}$ can be decomposed according to $$A^T=UB^T\in\mathbb R^{n\times m}\tag1$$ for some left-orthogonal (...
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### Does $A^TA=I$ imply $AA^T=I$?

Let $m,n\in\mathbb R^{m\times n}$ with $$A^TA=I_n\tag1.$$ I wonder whether this implies that $$AA^T=I_m\tag2$$ or if we can show it at least in the case $m=n$. EDIT: I was hoping for a proof which ...
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### Form a new base with two vectors

I need to make a new orthonormal base in $\mathbb{R^3}$ given $(2,7,5)$ and $(4,1,3)$ so that it makes $(\widehat{e_1}, \widehat{e_2}, \widehat{e_3} )$. But $\widehat{e_1}$ has the same direction of ...
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### Prove (a ∧ b) · c does NOT equal 0 if and only if a, b and c are linearly independent

Prove that $(a ∧ b)\cdot c\ne0\iff a, b$ and $c$ are linearly independent, where $a,b,c\in\Bbb R^3$. In the first part of the question, I proved that $(a ∧ b)$ is orthogonal to $a$ and to $b$, ...
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### How to prove polarisation identity?

Given that $A$ is a complex $n\times n$ Matrix. How do I prove that $\langle A(u + v) , u + v\rangle − \langle A(u − v), u − v \rangle = 2\langle Au, v \rangle + 2\langle Av, u \rangle$ ? I am stuck ...
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### Rayleigh quotient and orthogonality.

I'm studying chapter 6 of David Lay's Linear algebra and Its Applications and I'm having a little trouble understanding what this exercise has to do with orthogonality. The following has been ...
In electrodynamics I have seen the following: Let $\phi$ be a solution to the Poisson equation $-\Delta \phi= \rho$, and assume that $\rho$ is compactly supported. Then we can expand $\phi$ as the ...
Given a basis of complex eigenvectors for, say, a $2 \times 2$ symmetric matrix $A$ (which hence has real eigenvalues). Can one generate a basis of real(-valued) eigenvectors from the real and ...