# Tagged Questions

An inner product space is a vector space equipped with an inner product. The inner product is a generalisation of the "dot" product often used in vector calculus.

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### Inner Product of Imaginary Component and Real Component of an Orthogonal Function

I would like to evaluate the inner product of two components of a basis function that is part of a complete set. The two components are the imaginary and real components of the basis function. Could I ...
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### Proportinal line elements imply preservations of angles.

Consider a Riemannian manifold $(M,g)$, and a variation of the line element $\delta ds^2$ that is proportional to the original line element $ds^2$. This is $\delta ds^2=c ds^2$ for some constant $c$. ...
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### Matrix representing inner product?

Is is true that given an inner product space $\left\langle -,- \right\rangle$, there's a matrix $A$ such that $\left\langle u,v \right\rangle =\left\langle Au,Av \right\rangle _\mathbb{R^n}$? I ...
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### Projections satisfying $\| Px-Qx \| <\|x \|$ for nonzero $x$?

Let $V$ be a f.d inner product space with subspace $M,N$ and corresponding orthogonal projections $Q,P$. I need to prove that if $\| Px-Qx \| <\|x \|$ for all nonzero $x$, then $\dim M=\dim N$. As ...
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### How can find the vector that satisfy some conditions

I have a question Assume that there are 3 vectors x1,x2,x3 (each vector has the size 3*1 (3 dimension)) I want to find these vector that satisfy below conditions (the ininitial assumption x1 = [1 0 ...
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### If $\ \|v\|^{2}=\sum \langle v | v_{i} \rangle^2$ for every $v$ then the set $\{v_{i}\}$ is orthogonal

Suppose that $\{v_{1}\cdot\cdot\cdot v_{n}\}$ are unit vectors in $\mathbb{R}^{n}$ such that $$\ \|v\|^{2}=\sum_{1}^{n} |\langle v | v_{i} \rangle|^{2}, \forall v\in\mathbb{R}^{n}.$$ Then how to ...
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### Expressing the orthogonal projections on a linear operator $T$'s eigenspaces as polynomials in $T$

In the inner product space $\mathbb{C}^{2}$ with its standard inner product, let $$T\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3x+4y\\-4x+3y \end{pmatrix}$$ a linear operator. Express the ...
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### Show that following three statements are equivalent

Proposition: Let $V$ be finite-dimensional inner-product space and $A\in L(V)$. Show that following three statements are equivalent:1) A is hermitian operator.2) For every orthonormal basis $b$ in $V$ ...
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### If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
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