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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41 views

Finding an orthonormal basis for the plane $x_1 - 5x_2 - x_3 = 0$

Find an orthonormal basis of the plane $x_1 - 5x_2 - x_3 = 0$ I'm having trouble with this problem. So I picked the vectors $u_1 = \begin{bmatrix}1\\0\\1\end{bmatrix}$ and $u_2 = \begin{bmatrix}5\\...
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2answers
48 views

Let $U$ be an $m \times n$ matrix where the columns of $U$ form an orthonormal set

Let $U$ be an $m \times n$ matrix where the columns of $U$ form an orthonormal set. a) If $\vec{x}$ and $\vec{y}$ are in $\mathbb{R}^n$, show that $(U\vec{x})*(U\vec{y}) = \vec{x} * \vec{y}$ ...
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18 views

Finding an approximate function using orthonormal basis

I'm trying to take a function in $C_0[0,1]$ space (let's call this $f(x)$) and trying to find the best approximate of $f(x)$ at $P_2[0,1]$ space (let's call this approximate $p(x)$). Note that $P_2[0,...
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3answers
36 views

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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0answers
17 views

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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1answer
26 views

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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22 views

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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2answers
43 views

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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1answer
46 views

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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1answer
57 views

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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0answers
15 views

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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2answers
52 views

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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1answer
37 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
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0answers
33 views

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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2answers
69 views

For an orthogonal matrix $Q$, why does $QQ^T = I$?

In my linear algebra text (Strang), an orthogonal matrix is defined to be a square matrix whose columns are orthonormal. In other words, an orthogonal matrix is a matrix $Q = [q_1 \cdots q_n]$ where ...
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1answer
40 views

Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
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1answer
21 views

Orthonormal set of basis functions in $L^2([a,b])$

I am wondering about the basis functions for $$e_n(x)=\frac{e^{2\pi i n x/L}}{\sqrt(L)}$$ where $L = b - a$ on the domain of $L^2([a,b])$ when doing fourier series. Basically, we must scale it by $L = ...
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1answer
49 views

the proof of variational principal for the principal eigenvalue (checking orthonormal subset)

Hi I am looking at part 3 of the proof in Evans Chapter 6. I have difficulty understanding "Furthermore from (6) and (7) we see that $(\lambda_k^{-1/2} w_k)$ is an orthonormal subset of $H_0^1(U)$. ...
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1answer
80 views

Orthonormal basis with specific norm

I need to show that $v_1,...,v_n$ is basis for $V$ whenever $e_1,...,e_n$ is an orthonormal basis for V and $v_1,...,v_n$ are vectors in $V$ such that $$\left\Vert e_i-v_i\right\Vert < \frac{1}{\...
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1answer
53 views

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 ...
2
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2answers
44 views

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 ...
2
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3answers
51 views

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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1answer
30 views

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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1answer
36 views

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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4answers
68 views

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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25 views

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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26 views

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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19 views

If $U_0,V$ are Hilbert spaces, $(e_n)$ is an ONB of $U_0$ and $ι:U_0→V$ is an embedding, can we complete $(ιe_n)$ to an ONB of $V$?

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }...
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15 views

Derive the relations between cartesian co-cordinate systems.

Let $K$ and $\bar K$ be two cartesian co-orditate systems in $\mathbb{R}^3$. The element: $$s^2=(\Delta x^1)^2+(\Delta x^2)^2+(\Delta x^3)^2$$ is an invariant in all co-ordinate system. I want prove ...
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1answer
29 views

Find an orthogonal basis for $\mathbb P_2$

The problem: For polynomials $\mathbb{P_2}$ we define the inner product between p and q as: $$ \langle p,q\rangle =p(t_0)q(t_0)+p(t_1)q(t_1)+p(t_2)q(t_2) $$ with $$t_0=0, t_1=1, \textrm{ and } ...
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2answers
23 views

Finding orthogonal operators in $\mathbb{R}^2$

Let $V = \mathbb{R}^2$ where $V$ is an inner product space with dot product. Let $v \in V$ be a unit vector. I want to show there are exactly two orthogonal operators $T: V \to V$ such that $T((1,0)) ...
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2answers
73 views

Eigenvalues and eigenvectors of the Householder matrix $H = I - \frac{2}{u^Tu} uu^T$

So during my first revision for the semester exams, I went through exercises in books/internet and I found 2-3 that caught my eye. One of them was the following: Let $u \in \mathbb R^n$ be a non-...
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22 views

Orthonormal basis and matrix.

Let $E = \{e_1,_{...},e_n\} , F = \{f_1,_{...},f_n\} $ be 2 orthonormal basis of V. Does $[Id]_F^E $ the transfer matrix from E to F is orthogonal ? I think that the answer is yes, thought of shoving ...
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1answer
22 views

The signature of an inner product space does not depend on its basis

In R.W.R. Darling's "Differential Forms and Connections" an inner product is defined for a vector space $V$ as a bilinear, symmetric and nondegenerate (but not necessarily positive-definite) map from $...
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37 views

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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16 views

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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15 views

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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1answer
23 views

Finding an orthonormal basis for subspace intersection, $W^{\perp}\cap V$

I am stuck trying to find an orthonormal basis for $W^{\perp}\cap V$. I'm given V = span$\{v_1,v_2,v_3\}$ and that $$ v_1= \begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 ...
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1answer
50 views

Generalized Hermite Function as eigenfunction of a differential operator

I'm going through this paper. The article defines function function $\phi_n^\mu(x)$ that is orthonormal on $L^2$ with measure $dm = dx$: \begin{equation} \phi^\mu_n =\left(\frac{\gamma_\mu(n)}{\...
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1answer
16 views

Find orthonormal bases for the orthogonal subset

Let S be the subset $$\{\left(1,0,2\right),\left(3,2,1\right),\left(1,-2,7\right)\}\subset\mathbb{R}^{3}$$ Find orthonormal bases for $S^{\perp}$ and $S^{\perp\perp}$ I have begun by putting these ...
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1answer
16 views

square eigenvectors for Singular Value Decomposition?

This is from my textbook What I don't understand is, $V$ and $U$ are already square, why the textbook says "if we want to make them square"?
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20 views

How to find a specific orthogonal basis for a hermitian Matrix

I got to show that every hermitian form has an orthognal basis. More precisely, that there exists a basis, so that the matrix of the hermitian form looks like $$ \begin{pmatrix} E_p & & \\ ...
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1answer
27 views

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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32 views

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 $Av_1 = u_1, . . . Av_n = u_n$.
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33 views

Gram-Schmidt Process, finding orthonormal basis

Suppose I'm given $2$ random vectors $(v_1,v_2)$ I want to find orthonormal basis $(w_1,w_2)$ Are the following equivalent? for the $w_2$ case $$w_1=\frac{v_1}{\|v_1\|}$$ $$x_2=v_2-\frac{(v_1,v_2)}{...
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4answers
46 views

How do I find orthonormal basis of $U$?

Let $U$ be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find orthonormal basis of $U$?
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2answers
32 views

Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis for some inner product.

Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis with respect to some inner product. How would i go about doing this?
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0answers
18 views

Point in $U$ closest to $(2,0,1,6)$

Let $U$ be the intersection of $x_1+x_2+x_3+x_4=0$ and $x_1+2x_2+3x_3+4x_4=0$ in $\mathbb{R}^4$. The basis of $U$ is $((1,-2,1,0), (2,-3,0,1))$ and the basis of the orthogonal space $U^\perp$ is $((3,...
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2answers
42 views

Write a vector as a linear combination of orthonormal set vectors

How can I determine whether a vector can be expressed as a linear combination of a orthonormal set vectors ? Let's say I have a orthonormal set of vectors $\{v_1, v_2\}$: $$ v_1=\left(-\frac{1}{2},-\...
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1answer
22 views

Orthonormal Basis and Orthonormal Set

I was pretty confused about these two terms: 'orthonormal basis' and 'orthonormal set'. I know the orthonormal basis is just the normalized vectors of the orthogonal basis, but what about the ...