# 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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### Show that $|\det (a_1, \ldots, a_k, b_1, \ldots, b_{n-k})|=\left|\det\left(\left\langle a_i, d_j\right\rangle \right)\right|$

Let $U$, $L$ be subspaces such that $U \oplus L= \mathbb{R}^n$, and choose orthonormal bases $a_1,\dots,a_k$ for $U$, $b_1,\dots,b_{n-k}$ for $L$, and $d_1,\dots, d_{k}$ for $L^\bot$. I want to show ...
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### Concept of signal size for energy saving in optimization

I've been trying to redo this optimization problem from this paper, but on GEKKO Python code instead of MatLAB as they did, which is about finding the maximum Biodiesel concentration at final time: J =...
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### Orthonormal basis for complex vector space

I have to answer the question with true or false: Every complex vector space with an inner dot product has an orthonormal basis. I think it is false for the case $\dim V= \infty$. But i cant find a ...
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My professor during the class mentioned (page 35) that for two vectors in high-dimensional space (say dimension $d$), we typically have $$\left|\boldsymbol{u}^T \boldsymbol{v}\right| \approx \frac{\|\... 0 votes 1 answer 82 views ### Riesz Representation Theorem & Weak closure of orthonormal basis in Hilbert space [closed] So here it was shown in an infinite dimensional seperable Hilbert space H, with the orthonormal basis E=\{e_1,e_2,\dots \} that 0\in \overline{E}^w, where the right hand side denotes the weak ... 0 votes 1 answer 72 views ### Prove that which is an orthonomal basis of L^2(R) Prove$$\left \{\frac{1}{π^{\frac{1}{2}}}\left (\frac{i-x}{i＋x} \right )^n\frac{1}{i＋x} \right \}_{n＝-\infty }^{n＝\infty}is an orthonomal basis of $L^2(R)$ I tried something like taking the inner ...
This is a follow-up of this post. Here is the statement we want to prove. Let $\{x_j\}$ be an arbitrarily indexed orthonormal set in a Hilbert space(possibly non-separable). Show that the closed ...