# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\underline b$?
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### 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 ...
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### $A$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...