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Download. Free Animated Stickers for Microsoft Movie Maker Stick Figure. animated stickers to your screen using Microsoft Movie Maker.. Animated video is great for.Q:

How to prove that matrix product is an isometry?

Let $F:\ell^2\rightarrow\ell^2$ be defined by $F(x)=\left[\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right]$ and $G(\left[\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right])=\left[\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right]$

Let $T:\ell^2\rightarrow\ell^2$ be defined by $T(x)=F(G(x))$.Show that $T$ is an isometry.

I tried by writing $$T(x)=\left[\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right]$$ $$=\left[\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right]$$ $$=F\left(\left[\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right]\right)$$
$$=\left[\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right]$$
$$=x$$
Thus
$$\lVert T(x)\rVert=\lVert x\rVert$$
This just shows that $T(x)=x$ how can I show that it is an isometry?

A:

Hint: Show that $T$ is $0$ on $(\mathbb{R}^n)^\perp$ and is injective. For example, suppose $Tx=0$. Do you see why there can be no $y
e 0$ such that $y\perp\Bbb R^n$ and $y\perp x$? (Try

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