Here's a homework question:
Let ${u_1, \ldots, u_n}$ be an ONB in $C^n$. Assuming that $n$ is even, compute
$||u_1 - u_2 + u_3 -\cdots - u_n||$
I have no idea how to solve this. Can anyone help?
Thanks.
Here's a homework question:
Let ${u_1, \ldots, u_n}$ be an ONB in $C^n$. Assuming that $n$ is even, compute
$||u_1 - u_2 + u_3 -\cdots - u_n||$
I have no idea how to solve this. Can anyone help?
Thanks.
Let $n=2p$. We have \begin{align*} \lVert u_1-u_2+\cdots -u_n\rVert^2 &=\lVert \sum_{j=1}^pu_{2j-1}-u_{2j}\rVert^2 \\ &=\sum_{j=1}^p\lVert u_{2j-1}-u_{2j}\rVert^2\\ &= \sum_{j=1}^p 2=2p, \end{align*} so $\lVert u_1-u_2+\cdots -u_n\rVert=\sqrt n$.
You could do this by expanding out $\|x\|^2$, or perhaps you have seen a formula for $\|\sum_j c_j u_j\|$.