3
votes
1answer
82 views
+50

Showing a certain sequence is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$

The problem is to show $$\left\{\frac{1}{\pi^{1/2}(i+z)}\left(\frac{i-z}{i+z}\right)^n\right\}_{n=1}^{\infty}$$ is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$ In another exercise, I have ...
1
vote
2answers
38 views

Orthonormals, v1,v2.

Given: $2$ vectors $v1,v2 \ne 0$ and $v1 \ne v2$ and $v1,v2 \in R^n$ Prove: $\{v1\}^\bot = \{v2\}^\bot$ if and only if ${v_1,v_2}$ are linearly dependent. Well, I do have a solution for this but I ...
1
vote
2answers
24 views

Given a symmetric matrix A, find an orthogonal matrix S such that $S^TAS$ is a diagonal matrix

Given the symmetric matrix: $$A = \left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right)$$ find an orthogonal matrix $S$ such that $S^TAS$ is a ...
0
votes
2answers
20 views

Orthogonals,Orthonormals,bases.

Given: $B = ((2/3, 1/3,2/3),(1/3,2/3,-2/3),(2/3,-2/3,-1/3))$ How can I check if B is an orthonormal base of $R^3$? I thought about first checking if these vectors are a base. Then checking if they ...
0
votes
2answers
24 views

How to prove perpendicular vectors problem

If I have: $$|\underline a + \underline b| = |\underline a - \underline b|$$ how do I prove that $\underline a$ is perpendicular to $\underline b$?
3
votes
0answers
89 views

Question about orthogonal transformation / orthogonal matrices

I have a question about orthogonal transformations. If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product ...
2
votes
3answers
108 views

$A$ is a symmetric real matrix. Show that there is $B$ such that $B^3=A$

I'm having trouble with this question, I'd like someone to point me in the right direction. let $A$ be a n by n matrix with real values. show that there is another n by n real matrix $B$ such that ...
-1
votes
1answer
56 views

Find conjugate transpose of linear transform

A difficult question I've been trying to tackle but I seem to hit a dead end. let $V$ be an inner product space over $\mathbb R$. We are required to find $T^{*}$ such that $<T(u),v> = ...
0
votes
0answers
40 views

How do I normalize these vectors? Gram-Schmidt simple question

I was asked the following assignment: In the space $\mathbb R_3[X]$ with the inner product: $<f,g> = \int_{-1}^1 f(x)g(x)dx$, perform the gram-schmidt process to the basis: $1,x,x^2,x^3$. My ...
1
vote
2answers
85 views

Linear Algebra: vectors and orthonormal basis

$\text{(a)}$ Let $\mathbf{u}=[1\,\,2\,\,3\,\,1]^T$ and $\mathbf v=[0\,\,0\,\,1\,\,-3]^T$ in $\mathbb{R}^4$. Find an orthonormal basis for the space $\mathcal W$ such that $\mathbf w \,\bot\, ...
0
votes
0answers
46 views

Finding Appropriate Linearly Independent Set

V is a dimension n inner product space over R. Q is a positive definite, self adjoint linear map and is invertable. I have found an S such that $S^2=Q$. I have also show that, for a self adjoint ...
1
vote
1answer
92 views

If $\{u,v\}$ is an orthonormal set in an inner product space, then find $\lVert 6u-8v\rVert$

If $\{u,v\}$ is an orthonormal set in an inner product space, then find $\|6u-8v\|$. That's pretty much it. I'm trying to study for a quiz and can't figure it out. There are no examples in my book to ...
5
votes
3answers
109 views

No idea how to prove this property about symmetric matrices

This is from homework, so please hints only. Suppose $A$ is symmetric such that all of its eigenvalues are 1 or -1. Prove that $A$ is orthogonal. The converse is really easy, but I really have ...
1
vote
2answers
196 views

Proving that a set is an orthonormal basis

Any ideas on how to quickly show that $$ \left( \frac{1}{\sqrt{2\pi}}, \frac{\sin(x)}{\sqrt{\pi}}, \frac{\sin(2x)}{\sqrt{\pi}}, ..., \frac{\sin(nx)}{\sqrt{\pi}}, \frac{\cos(x)}{\sqrt{\pi}}, ...
1
vote
1answer
34 views

“Normalizing” a Function

One of our homework problems for my linear algebra course this week is: On $\mathcal{P}_2(\mathbb{R})$, consider the inner product given by $$ \left<p, q\right> = \int_0^1 p(x)q(x) \, dx $$ ...
0
votes
0answers
40 views

tight frame for $\mathbb{C}^N$

I have a question to ask Prove that if $K\in\mathbb{Z}-\{0\}$, then $\{\phi_p[n]=\exp(i2\pi pn/(KN))\}_{0\leq p<KN}$ is a tight frame of $\mathbb{C}^N$, i.e. $\sum_{k}|\langle f,\phi_p\rangle ...
1
vote
1answer
36 views

Show that ON-sequence is a base

I have a Hilbert space $H$ and a base $(e_n)_{n=1}^\infty$ and a ON-sequence $(f_n)_{n=1}^\infty$. Given $$ \sum_{n=1}^\infty ||e_n - f_n||^2 < 1 $$ show that $(f_n)_{n=1}^\infty$ is a base. My ...
1
vote
1answer
82 views

Given a positive definite matrix $A$, how can I find a matrix $S$ such that $A=S^TS$ and $S$ has following restrictions?

My former question on this problem was not that popular, I was asking to prove the existence and uniqueness of a upper triangular matrix $S$ with positive entries on diagonal such that $A=S^TS$ for a ...
1
vote
1answer
70 views

Prove that for an Hermitian matrix $A$ exists one and only matrix $S\in \text{GL}(\mathbb{C},n)$ such that …

The task is to prove that for an Hermitian matrix $A$ exists one and only matrix $S\in \text{GL}(\mathbb{C},n)$ upper triangular with positive entries on diagonal such that $A = S^T \bar{S}$? As I ...
2
votes
1answer
41 views

What is wrong with my approach for orthonormal procedure?

I am given a matrix $$A=\left(\begin{matrix}1&1&0&1\\1&2&1&0\\0&1&3&-2\\1&0&-2&4\end{matrix}\right)$$ which defines a scalar product in $\mathbb{R}^4$ ...
1
vote
1answer
24 views

A Problem with a Proof Concerning the Properties of Orthogonal Matricies

OK, I'm working on the following problem and don't understand how my Linear Algebra text has made it to a certain conclusion here is the problem: Let u be a unit vector in $R^{n}$ and let $H = I - ...
1
vote
1answer
312 views

Haar basis on $L^2(0,1)$ - proof?

I have the following problem. We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$, $$f_{j,n}(t)=\left\{ ...
2
votes
1answer
59 views

Why is $\mathbf{\hat{x}} = \sum_{j=2}^n (\mathbf{x}\cdot\mathbf{u}_j)\mathbf{u}_j = \mathbf{x} - (\mathbf{x}\cdot \mathbf{u})\mathbf{u}$?

Let $\mathbf{u}, \mathbf{u}_2,\dots,\mathbf{u}_n$ be an orthonormal basis for $\mathbb{R}^n$, and let $W = Span\{\mathbf{u}_2, \dots, \mathbf{u}_n\}$ . Then the projection of $\mathbf{x}$ onto $W$ is ...
4
votes
2answers
2k views

Orthonormal Eigenbasis

I am a little apprehensive to ask this question because I have a feeling it's a "duh" question but I guess that's the beauty of sites like this (anonymity): I need to find an orthonormal eigenbasis ...