# Tagged Questions

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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### Smallest possible value of the norm?

The vectors $\vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix}$ and $\vec{u_2} = \begin{bmatrix} 1 \\ -1 \\ 1\\ -1 \end{bmatrix}$ are orthonormal in $\mathbb{R}^4$....
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### Relation between row-space and column-space vectors

Let $A$ be any $n$ by $m$ matrix. $V$ is an orthonormal vector in column-space of $A$. $U$ is an orthonormal vector in row-space of $A$. Now, why is the following relation True? $$AV=U\Sigma$$ , ...
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### Orthonormal basis in inner product space

Given that $V$ is an inner product space and ${e_n} \in V ,n=1,2,3...$ is an Orthonormal basis and $\lambda_n$ is a scalar series, How did they get from the first step to the second step?
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### How to find the matrix of orthonormal base vectors of a hyperplane?

How to find the matrix of orthonormal base vectors of a hyperplane translated such that it crosses the origin of parameter space? for example this hyperplane: $p1 + p2 = 1$ in a 3D space with this ...
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### Can the product of two orthonormal matrices be an orthonormal matrix?

Suppose you have a real orthonormal matrix $L$. Are there any real orthonormal matrices $X$, other than $L'$ and the identity matrix such that $Y=LX$ is also an orthonormal matrix?
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### Number of Eigenvectors in a Symmetric Matrix

Supposing symmetric matrix $A_{n\times n}$, how do I know that there are $n$ eigenvectors of $A$? By way of trying to communicate context, I've spent a rather unproductive 5 or so hours watching ...
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### How is the study of wavelets not just a special case of Fourier analysis?

As far as I can tell, "wavelets" is just a neologism for certain "non-smooth" families of functions which constitute orthonormal bases/families for $L^2[0,1]$. How is wavelet analysis anything new ...
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### Given A $m\times n$, $\{u_1,…,u_n\}$ ON basis of $R^n$, prove eigenvalues aren't negative [duplicate]

Given $A$ $(m\times n)$, $\{u_1,...,u_n\}$ ON(orthonormal) basis of $R^n$ which are eigenvectors of $A^TA$ with $\lambda_1 , ... , \lambda_n$ eigenvalues accordingly. Prove: Eigenvalues are not ...
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### Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$ Basically if I have found the orthonormal basis for the span of S can I use that to find the dimension ...
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### How can we compute the square root of an operator of the form $Cv=\sum_{n\in\mathbb N}\langle v,e_n\rangle_Ve_n$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $V$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ and that the inclusion $\iota$ is Hilbert-Schmidt $C:=\iota^\ast$ denote the ...
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### Endomorphism is normal and idempotent iff it is an orthogonal projection.

I've searched for answers for this question here for some time but haven't found an applicable answer because I could only find related questions, but not this one in particular. Suppose $V$ is a ...
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### Rule for $\langle x,y\rangle$ if we know orthonormal base?

How to define $\langle x,y \rangle$ in space of polinoms, where $1, x-1 , 1-x^2$ are orthonormal base($\Vert a\Vert = 1$, $\langle a1, a2\rangle = 0$)? I'm a bit lost, I know how to do it with my ...
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### Riesz Representation Thereom for Polynomials with real coefficients problem

Find a polynomial q(x) $\in$ P$_2$($\Bbb R$) Such that $p ( 1/4 ) = $$\int_0^1 p(x)q(x) \,dx$$$. I'm sorry to ask this question, but I've been working on it for some time. The inner product on P$_2$(...
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### Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
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### Let the column vectors of a $3 × 3$ matrix $A$ form an orthonormal basis. Explain why $A^T = A^{−1}$ .

Let the column vectors of a $3 × 3$ matrix $A$ form an orthonormal basis. Explain why $A^T = A^{−1}$ . My Attempt: $AA^T=I$ if and only if $A^TA=I$. So $A$ is orthogonal if and only if $A^T$ is ...
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### Find the orthonormal projection?

I am trying to find the orthonormal projectiont of $X\in V\equiv M_{2\times2}(\mathbb{R})$ over the space of diagonal matrix defined by $$D=\{M=(m_{ij})\in V:m_{ij}=0,\ i\not=j\}\ \ \ i,j\in\{1,2\}$$ ...
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### Transformation matrix is jordan normal form

I have the following question: Given a finite-dimensional, unitary vector space V and a endomorphism f on V, is it possible to choose an orthonormal basis B of V in such a way, that the transformation ...
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### weak solution via Galerkin approximation

I am reading Evan's approach to Existence of weak solution via Galerkin method in some PDE notes. I have difficulty understand the following assumption i.e. Let "$\{w_k\}_{k=1}^\infty$ be an ...
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### for every linear map $\ T:V\to V$ : $\ [T^*]_B=(M^t)^{-1}A^tM\$ when $\ [T]_B=A$.

Let $V$ be an inner product space of finite dimension over $\mathbb{R}$ and Let $B=\{v_1,...v_n\}$ be a basis of V (not necessarily orthonormal). Let $M\in M_n(\mathbb{R})$ a matrix whose i,j element ...
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### Extending $\{u_1, u_2\}$ to an orthonormal basis when finding an SVD

I've been working through my linear algebra textbook, and when finding an SVD there's just one thing I don't understand. For example, finding an SVD for the 3x2 matrix A. I will skip the steps of ...
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### Rank of product of SVD vectors

Given a matrix $M$, we can compute its singular value decomposition $M=U\Sigma V^*$ where $^*$ is the complex conjugate transpose. $U$ and $V$ are unitary, so $UU^*=I$, $VV^*=I$. Let's take the $i$-th ...
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### Transformation of Orthonormal Bases

Suppose that $u_1, . . . , u_n$ and $v_1, . . . , v_n$ are orthonormal bases for $\Bbb{R}^n$. Construct the matrix A that transforms each $u_i$ into $v_i$ to give $Av_1 = u_1, . . . Av_n = u_n$.