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I have a previous post here. There is a part b to that question and it asks:
Let $x=(1,1,1)^T$. Write x as a linear combination of $u_1, u_2, u_3$ using Parseval's formula to compute $||x||$.

I know how to compute $||x||$, it's simply the magnitude. However I am totally unsure how do to x as a linear combo, I've read through my book and tried looking online with no luck. Any ideas?

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Hint: Since you have orthonormal basis $\{u_1,u_2,u_3\}$, you can express any vector $x$ as a linear combination of the basis vectors as $x=\alpha_1 u_1+\alpha_2 u_2+\alpha_3 u_3$ where you have to determine the coefficients (scalars) $\alpha_i$'s.

Since, $u_i$'s are orthonormal, $=u_i^t.u_j=\delta_{ij}$ (=1, only if $i=j$, else $0$). So,

$x^t.u_i==(\sum\alpha_j u_j^t).u_i=\alpha_i$.

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    Excellent! You are welcome.2012-10-12