# 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.

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### Is a linear operator $T$ that is normal, normal regardless of the ordered basis while the matrix representation of that operator must be on an ortho..

Is a linear operator $T$ that is normal, normal regardless of the ordered basis while the matrix representation of that operator must be on an orthonormal basis? I was attempting an exercise that ...
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### Questions about the eigenfunctions and eigenvalues of the momentum operator $\hat{p}$

I'm studying Quantum Mechanics right now and working through an example in the book of an eigenfunction with a continuous spectrum - the momentum operator, $\hat{p} = -i\hbar\frac{d}{dx}$. In the ...
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### Axler 6.9: Show that the following list is orthonormal.

Let $n \in \mathbb{Z_{+}}$ and show that the list below is an orthonormal list of vectors in $C[-\pi,\pi]$ in the vector space of real valued functions on $[-\pi,\pi]$ with the inner product given ...
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### Absolute value of the determinant of a linear map between two equal dimensional Euclidean vector spaces is independent of orthonormal basis

Let $V_1$ and $V_2$ be two Euclidean vector spaces with the same dimension, and let $f : V_1 \longrightarrow V_2$ be a linear map between them. If $M$ is the matrix of the map $f$ relative to two ...
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### If a linear map sends orthonormal basis on orthonotmal basis then it is an isometry?

Let $(\mathbf{u_1,u_2,u_3})$ and $(\mathbf{v_1,v_2,v_3})$ orthonormal lists. Define $T:\mathbb{R^3}\rightarrow \mathbb{R^3}$ through the lineal extension $T(u_k)=v_k$. Is T an isometry? My attempt: I ...
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### Show $Q(x)\cdot Q(y)=x\cdot y\Rightarrow Q^TQ=I$

According to my class notes the following are two equivalent definitions of an orthogonal matrix: $Q^TQ=I$ $Q(x)\cdot Q(y)=x\cdot y$ I've been able to show that $1\Rightarrow 2$, yet I do not know ...
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### Frobenius norm equality related to "orthonormal pair"

Related to this question as far as concerned the paper discussed. In the following theorem the author is not very clear about who $(W,W-)$ and $M$ should be. Morover, the notation orthonormal pair I'm ...
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Here is the full exercise: From my understanding I need to show: $||\varphi_p||^2=\frac{1}{\Delta x}\sum_{j=0}^{M}\sin^2(\pi pj\Delta x)=1$ $p \neq q \implies (\varphi_p,\varphi_q)=0$ $\... 1 vote 0 answers 35 views ### Proving that a bounded linear operator$A \in \mathscr{I}_1\$

Can someone please help me with the following problem? I have some of my work below but I am not sure if I attacked the problem wisely. Thank you for your time and consideration. Suppose that a ...
Consider this post. Questions: For equation (2) from Abel, did the person mean to say $$b - \alpha q_1 - \color{red}\beta q_2 = (q_1^Tb - \alpha)q_1 + (q_2^Tb-\beta)q_2 + \epsilon?$$ Working out the ...