# Showing a given set of vectors forms a Parseval frame

1. Feb 6, 2013

### SithsNGiggles

1. The problem statement, all variables and given/known data

Show that the vectors
$\sqrt{\frac{2}{3}}(1,0), \sqrt{\frac{2}{3}}\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right), \sqrt{\frac{2}{3}}\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$
form a Parseval frame of $\mathbb{R}^2$, but are neither linearly independent nor orthonormal

2. Relevant equations

The definition of Parseval frame, according to class notes, is
"A sequence of vectors $\displaystyle\left\{x_i \right\}_{i=1}^{k}$ of an inner product space $V$ of dimension $n (n\leq k)$ is called a Parseval frame for $V$ if $\forall x\in V$,
$||x||^2=\displaystyle\sum_{i=1}^{k}|\langle x,x_i \rangle|^2$.​

3. The attempt at a solution

I'm not quite sure how to interpret the definition. Or maybe I do, I just don't know how to implement it.

What I've got so far:
$||x||^2 = \langle x,x\rangle = \displaystyle\sum_{i=1}^{3}|\langle x,x_i \rangle|^2$
$||x||^2 = \left|\left\langle x,\sqrt{\frac{2}{3}}(1,0)\right\rangle\right|^2 + \left|\left\langle x,\sqrt{\frac{2}{3}}\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2 + \left|\left\langle x,\sqrt{\frac{2}{3}}\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2$

$||x||^2 = \frac{2}{3} \left[\left|\left\langle x,(1,0)\right\rangle\right|^2 + \left|\left\langle x,\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2 + \left|\left\langle x,\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2\right]$

Then, I suppose I take the dot product since $\mathbb{R}^2$ is my inner product space, so I let $x = (x_1,x_2)$, where $x_1,x_2\in\mathbb{R}$. Then I write
$||x||^2 = \frac{2}{3} \left[|x_1|^2 + \left| -\frac{1}{2}x_1 + \frac{\sqrt{3}}{2}x_2 \right|^2 + \left| -\frac{1}{2}x_1 - \frac{\sqrt{3}}{2}x_2 \right|^2\right]$

Factoring out some constants gives me
$||x||^2 = \frac{2}{3} \left[|x_1|^2 + \frac{1}{4}\left|x_1 - \sqrt{3}x_2 \right|^2 + \frac{1}{4}\left|x_1 + \sqrt{3}x_2 \right|^2\right]$

And that's all I've done. I'm not sure if I'm even doing this right. I've already shown that the vectors aren't linearly independent and orthonormal. Any ideas? Thanks.